Computation of equivariant multiplicities for $\mathrm{Hilb}^2$¶
In this worksheet, we compute equivariant multiplicities for $\mathrm{Hilb}^2(S)$, where $S$ is a 2d integrable system. We only concern ourselves with the hardest case, namely wobbly support, since the rest was treated explicitly in the paper. Unfortunately, the implementation relies on a lot of copy-pasting.
Local equations¶
Here, we compute the local equations of the preimage of origin under the map $\mathrm{Sym}^2(S)\to \mathrm{Sym}^2(\mathbb{C})$. As opposed to the paper, we denote $f$ by $p_1$, and $g$ by $p_2$.
In [43]:
#adapted from https://groups.google.com/g/sage-support/c/aJn0T0jIfwU
Var.<x1,x2,y1,y2,s,u,t,w,v> = PolynomialRing(QQ,order="degrevlex(4),degrevlex(5)")
II = Var.ideal([v^2 - u*w,s-x1-x2,u-(x1-x2)^2,t-y1-y2,w-(y1-y2)^2,v-(x1-x2)*(y1-y2)])
inv=lambda f:QQ['s,u,t,w,v']((II.reduce(f)))
def p1(k, l):
return 2^(k+l-1)*inv(x1^k*y1^l+x2^k*y2^l)
def p2(k, l):
return 2^(2*k+2*l)*inv((x1*x2)^k*(y1*y2)^l)
#the following prints the polynomials f=p1, g=p2 from the paper, up to a constant, in all cases of interest
print("xy²: p1 =",p1(1,2),"\n p2 =",p2(1,2),"\n--------------")
print("x²y³: p1 =",p1(2,3),"\n p2 =",p2(2,3),"\n--------------")
print("x²y⁴: p1 =",p1(2,4),"\n p2 =",p2(2,4),"\n--------------")
print("x³y⁴: p1 =",p1(3,4),"\n p2 =",p2(3,4),"\n--------------")
print("x⁴y⁵: p1 =",p1(4,5),"\n p2 =",p2(4,5),"\n--------------")
xy²: p1 = s*t^2 + s*w + 2*t*v p2 = s^2*t^4 - u*t^4 - 2*s^2*t^2*w + s^2*w^2 + 2*t^2*v^2 - w*v^2 -------------- x²y³: p1 = s^2*t^3 + u*t^3 + 3*s^2*t*w + 6*s*t^2*v + 2*s*w*v + 3*t*v^2 p2 = s^4*t^6 - 2*s^2*u*t^6 - 3*s^4*t^4*w + u^2*t^6 + 3*s^4*t^2*w^2 + 6*s^2*t^4*v^2 - s^4*w^3 - 3*u*t^4*v^2 - 6*s^2*t^2*w*v^2 + 2*s^2*w^2*v^2 + 3*t^2*v^4 - w*v^4 -------------- x²y⁴: p1 = s^2*t^4 + u*t^4 + 6*s^2*t^2*w + 8*s*t^3*v + s^2*w^2 + 8*s*t*w*v + 6*t^2*v^2 + w*v^2 p2 = s^4*t^8 - 2*s^2*u*t^8 - 4*s^4*t^6*w + u^2*t^8 + 6*s^4*t^4*w^2 + 8*s^2*t^6*v^2 - 4*s^4*t^2*w^3 - 4*u*t^6*v^2 - 12*s^2*t^4*w*v^2 + s^4*w^4 + 8*s^2*t^2*w^2*v^2 + 6*t^4*v^4 - 2*s^2*w^3*v^2 - 4*t^2*w*v^4 + w^2*v^4 -------------- x³y⁴: p1 = s^3*t^4 + 3*s*u*t^4 + 6*s^3*t^2*w + 12*s^2*t^3*v + s^3*w^2 + 4*u*t^3*v + 12*s^2*t*w*v + 18*s*t^2*v^2 + 3*s*w*v^2 + 4*t*v^3 p2 = s^6*t^8 - 3*s^4*u*t^8 - 4*s^6*t^6*w + 3*s^2*u^2*t^8 + 6*s^6*t^4*w^2 + 12*s^4*t^6*v^2 - u^3*t^8 - 4*s^6*t^2*w^3 - 12*s^2*u*t^6*v^2 - 18*s^4*t^4*w*v^2 + s^6*w^4 + 4*u^2*t^6*v^2 + 12*s^4*t^2*w^2*v^2 + 18*s^2*t^4*v^4 - 3*s^4*w^3*v^2 - 6*u*t^4*v^4 - 12*s^2*t^2*w*v^4 + 3*s^2*w^2*v^4 + 4*t^2*v^6 - w*v^6 -------------- x⁴y⁵: p1 = s^4*t^5 + 6*s^2*u*t^5 + 10*s^4*t^3*w + 20*s^3*t^4*v + u^2*t^5 + 5*s^4*t*w^2 + 20*s*u*t^4*v + 40*s^3*t^2*w*v + 60*s^2*t^3*v^2 + 4*s^3*w^2*v + 10*u*t^3*v^2 + 30*s^2*t*w*v^2 + 40*s*t^2*v^3 + 4*s*w*v^3 + 5*t*v^4 p2 = s^8*t^10 - 4*s^6*u*t^10 - 5*s^8*t^8*w + 6*s^4*u^2*t^10 + 10*s^8*t^6*w^2 + 20*s^6*t^8*v^2 - 4*s^2*u^3*t^10 - 10*s^8*t^4*w^3 - 30*s^4*u*t^8*v^2 - 40*s^6*t^6*w*v^2 + u^4*t^10 + 5*s^8*t^2*w^4 + 20*s^2*u^2*t^8*v^2 + 40*s^6*t^4*w^2*v^2 + 60*s^4*t^6*v^4 - s^8*w^5 - 5*u^3*t^8*v^2 - 20*s^6*t^2*w^3*v^2 - 40*s^2*u*t^6*v^4 - 60*s^4*t^4*w*v^4 + 4*s^6*w^4*v^2 + 10*u^2*t^6*v^4 + 30*s^4*t^2*w^2*v^4 + 40*s^2*t^4*v^6 - 6*s^4*w^3*v^4 - 10*u*t^4*v^6 - 20*s^2*t^2*w*v^6 + 4*s^2*w^2*v^6 + 5*t^2*v^8 - w*v^8 --------------
Equations after blowup¶
Here, we compute the local equations after blowup. The atlas $\psi_1$ is responsible for the neighborhood of the point $I_p^{(1^2)}$, and $\psi_2$ for the neighborhood of $I_p^{(2)}$. Recall that non-equivariant multiplicities are supposed to be $b^2$ and $2b^2$ respectively. For consistency, we change the notations somewhat compared to the paper. In the first atlas, we write $X_1=s, X_2=\eta, Y_1=t, Y_2=w_0$. In the second atlas, we write $X_1=s, X_2=u_0, Y_1=t, Y_2=\xi$.
In [44]:
S.<s,t,u,v,w> = QQ[]
U.<Y1,Y2,X1,X2> = PolynomialRing(QQ,order="negdeglex(2),negdeglex(2)")
psi1 = S.hom([U(X1),U(Y1),U(X2^2*Y2),U(X2*Y2),U(Y2)],U)
psi2 = S.hom([U(X1),U(Y1),U(X2),U(Y2*X2),U(Y2^2*X2)],U)
#the following prints the polynomials f=p1, g=p2 after substitutions in 2 atlases of blowup
#notations for 1st atlas: X1=s, X2=eta, Y1=t, Y2=w0
#notations for 2nd atlas: X1=s, X2=u0, Y1=t, Y2=xi
print("ψ₁(xy²): \np₁ =",psi1(p1(1,2)),"\np₂ =",psi1(p2(1,2)))
print("ψ₂(xy²): \np₁ =",psi2(p1(1,2)),"\np₂ =",psi2(p2(1,2)),"\n--------------")
print("ψ₁(x²y³): \np₁ =",psi1(p1(2,3)),"\np₂ =",psi1(p2(2,3)))
print("ψ₂(x²y³): \np₁ =",psi2(p1(2,3)),"\np₂ =",psi2(p2(2,3)),"\n--------------")
print("ψ₁(x²y⁴): \np₁ =",psi1(p1(2,4)),"\np₂ =",psi1(p2(2,4)))
print("ψ₂(x²y⁴): \np₁ =",psi2(p1(2,4)),"\np₂ =",psi2(p2(2,4)),"\n--------------")
print("ψ₁(x³y⁴): \np₁ =",psi1(p1(3,4)),"\np₂ =",psi1(p2(3,4)))
print("ψ₂(x³y⁴): \np₁ =",psi2(p1(3,4)),"\np₂ =",psi2(p2(3,4)),"\n--------------")
print("ψ₁(x⁴y⁵): \np₁ =",psi1(p1(4,5)),"\np₂ =",psi1(p2(4,5)))
print("ψ₂(x⁴y⁵): \np₁ =",psi2(p1(4,5)),"\np₂ =",psi2(p2(4,5)),"\n--------------")
ψ₁(xy²): p₁ = Y2*X1 + Y1^2*X1 + 2*Y1*Y2*X2 p₂ = Y2^2*X1^2 - 2*Y1^2*Y2*X1^2 - Y2^3*X2^2 + Y1^4*X1^2 + 2*Y1^2*Y2^2*X2^2 - Y1^4*Y2*X2^2 ψ₂(xy²): p₁ = Y1^2*X1 + 2*Y1*Y2*X2 + Y2^2*X1*X2 p₂ = -Y1^4*X2 + Y1^4*X1^2 + 2*Y1^2*Y2^2*X2^2 - 2*Y1^2*Y2^2*X1^2*X2 - Y2^4*X2^3 + Y2^4*X1^2*X2^2 -------------- ψ₁(x²y³): p₁ = 3*Y1*Y2*X1^2 + 2*Y2^2*X1*X2 + Y1^3*X1^2 + 6*Y1^2*Y2*X1*X2 + 3*Y1*Y2^2*X2^2 + Y1^3*Y2*X2^2 p₂ = -Y2^3*X1^4 + 3*Y1^2*Y2^2*X1^4 + 2*Y2^4*X1^2*X2^2 - 3*Y1^4*Y2*X1^4 - 6*Y1^2*Y2^3*X1^2*X2^2 - Y2^5*X2^4 + Y1^6*X1^4 + 6*Y1^4*Y2^2*X1^2*X2^2 + 3*Y1^2*Y2^4*X2^4 - 2*Y1^6*Y2*X1^2*X2^2 - 3*Y1^4*Y2^3*X2^4 + Y1^6*Y2^2*X2^4 ψ₂(x²y³): p₁ = Y1^3*X2 + Y1^3*X1^2 + 6*Y1^2*Y2*X1*X2 + 3*Y1*Y2^2*X2^2 + 3*Y1*Y2^2*X1^2*X2 + 2*Y2^3*X1*X2^2 p₂ = Y1^6*X2^2 - 2*Y1^6*X1^2*X2 + Y1^6*X1^4 - 3*Y1^4*Y2^2*X2^3 + 6*Y1^4*Y2^2*X1^2*X2^2 - 3*Y1^4*Y2^2*X1^4*X2 + 3*Y1^2*Y2^4*X2^4 - 6*Y1^2*Y2^4*X1^2*X2^3 + 3*Y1^2*Y2^4*X1^4*X2^2 - Y2^6*X2^5 + 2*Y2^6*X1^2*X2^4 - Y2^6*X1^4*X2^3 -------------- ψ₁(x²y⁴): p₁ = Y2^2*X1^2 + 6*Y1^2*Y2*X1^2 + 8*Y1*Y2^2*X1*X2 + Y2^3*X2^2 + Y1^4*X1^2 + 8*Y1^3*Y2*X1*X2 + 6*Y1^2*Y2^2*X2^2 + Y1^4*Y2*X2^2 p₂ = Y2^4*X1^4 - 4*Y1^2*Y2^3*X1^4 - 2*Y2^5*X1^2*X2^2 + 6*Y1^4*Y2^2*X1^4 + 8*Y1^2*Y2^4*X1^2*X2^2 + Y2^6*X2^4 - 4*Y1^6*Y2*X1^4 - 12*Y1^4*Y2^3*X1^2*X2^2 - 4*Y1^2*Y2^5*X2^4 + Y1^8*X1^4 + 8*Y1^6*Y2^2*X1^2*X2^2 + 6*Y1^4*Y2^4*X2^4 - 2*Y1^8*Y2*X1^2*X2^2 - 4*Y1^6*Y2^3*X2^4 + Y1^8*Y2^2*X2^4 ψ₂(x²y⁴): p₁ = Y1^4*X2 + Y1^4*X1^2 + 8*Y1^3*Y2*X1*X2 + 6*Y1^2*Y2^2*X2^2 + 6*Y1^2*Y2^2*X1^2*X2 + 8*Y1*Y2^3*X1*X2^2 + Y2^4*X2^3 + Y2^4*X1^2*X2^2 p₂ = Y1^8*X2^2 - 2*Y1^8*X1^2*X2 + Y1^8*X1^4 - 4*Y1^6*Y2^2*X2^3 + 8*Y1^6*Y2^2*X1^2*X2^2 - 4*Y1^6*Y2^2*X1^4*X2 + 6*Y1^4*Y2^4*X2^4 - 12*Y1^4*Y2^4*X1^2*X2^3 + 6*Y1^4*Y2^4*X1^4*X2^2 - 4*Y1^2*Y2^6*X2^5 + 8*Y1^2*Y2^6*X1^2*X2^4 - 4*Y1^2*Y2^6*X1^4*X2^3 + Y2^8*X2^6 - 2*Y2^8*X1^2*X2^5 + Y2^8*X1^4*X2^4 -------------- ψ₁(x³y⁴): p₁ = Y2^2*X1^3 + 6*Y1^2*Y2*X1^3 + 12*Y1*Y2^2*X1^2*X2 + 3*Y2^3*X1*X2^2 + Y1^4*X1^3 + 12*Y1^3*Y2*X1^2*X2 + 18*Y1^2*Y2^2*X1*X2^2 + 4*Y1*Y2^3*X2^3 + 3*Y1^4*Y2*X1*X2^2 + 4*Y1^3*Y2^2*X2^3 p₂ = Y2^4*X1^6 - 4*Y1^2*Y2^3*X1^6 - 3*Y2^5*X1^4*X2^2 + 6*Y1^4*Y2^2*X1^6 + 12*Y1^2*Y2^4*X1^4*X2^2 + 3*Y2^6*X1^2*X2^4 - 4*Y1^6*Y2*X1^6 - 18*Y1^4*Y2^3*X1^4*X2^2 - 12*Y1^2*Y2^5*X1^2*X2^4 - Y2^7*X2^6 + Y1^8*X1^6 + 12*Y1^6*Y2^2*X1^4*X2^2 + 18*Y1^4*Y2^4*X1^2*X2^4 + 4*Y1^2*Y2^6*X2^6 - 3*Y1^8*Y2*X1^4*X2^2 - 12*Y1^6*Y2^3*X1^2*X2^4 - 6*Y1^4*Y2^5*X2^6 + 3*Y1^8*Y2^2*X1^2*X2^4 + 4*Y1^6*Y2^4*X2^6 - Y1^8*Y2^3*X2^6 ψ₂(x³y⁴): p₁ = 3*Y1^4*X1*X2 + Y1^4*X1^3 + 4*Y1^3*Y2*X2^2 + 12*Y1^3*Y2*X1^2*X2 + 18*Y1^2*Y2^2*X1*X2^2 + 6*Y1^2*Y2^2*X1^3*X2 + 4*Y1*Y2^3*X2^3 + 12*Y1*Y2^3*X1^2*X2^2 + 3*Y2^4*X1*X2^3 + Y2^4*X1^3*X2^2 p₂ = -Y1^8*X2^3 + 3*Y1^8*X1^2*X2^2 - 3*Y1^8*X1^4*X2 + Y1^8*X1^6 + 4*Y1^6*Y2^2*X2^4 - 12*Y1^6*Y2^2*X1^2*X2^3 + 12*Y1^6*Y2^2*X1^4*X2^2 - 4*Y1^6*Y2^2*X1^6*X2 - 6*Y1^4*Y2^4*X2^5 + 18*Y1^4*Y2^4*X1^2*X2^4 - 18*Y1^4*Y2^4*X1^4*X2^3 + 6*Y1^4*Y2^4*X1^6*X2^2 + 4*Y1^2*Y2^6*X2^6 - 12*Y1^2*Y2^6*X1^2*X2^5 + 12*Y1^2*Y2^6*X1^4*X2^4 - 4*Y1^2*Y2^6*X1^6*X2^3 - Y2^8*X2^7 + 3*Y2^8*X1^2*X2^6 - 3*Y2^8*X1^4*X2^5 + Y2^8*X1^6*X2^4 -------------- ψ₁(x⁴y⁵): p₁ = 5*Y1*Y2^2*X1^4 + 4*Y2^3*X1^3*X2 + 10*Y1^3*Y2*X1^4 + 40*Y1^2*Y2^2*X1^3*X2 + 30*Y1*Y2^3*X1^2*X2^2 + 4*Y2^4*X1*X2^3 + Y1^5*X1^4 + 20*Y1^4*Y2*X1^3*X2 + 60*Y1^3*Y2^2*X1^2*X2^2 + 40*Y1^2*Y2^3*X1*X2^3 + 5*Y1*Y2^4*X2^4 + 6*Y1^5*Y2*X1^2*X2^2 + 20*Y1^4*Y2^2*X1*X2^3 + 10*Y1^3*Y2^3*X2^4 + Y1^5*Y2^2*X2^4 p₂ = -Y2^5*X1^8 + 5*Y1^2*Y2^4*X1^8 + 4*Y2^6*X1^6*X2^2 - 10*Y1^4*Y2^3*X1^8 - 20*Y1^2*Y2^5*X1^6*X2^2 - 6*Y2^7*X1^4*X2^4 + 10*Y1^6*Y2^2*X1^8 + 40*Y1^4*Y2^4*X1^6*X2^2 + 30*Y1^2*Y2^6*X1^4*X2^4 + 4*Y2^8*X1^2*X2^6 - 5*Y1^8*Y2*X1^8 - 40*Y1^6*Y2^3*X1^6*X2^2 - 60*Y1^4*Y2^5*X1^4*X2^4 - 20*Y1^2*Y2^7*X1^2*X2^6 - Y2^9*X2^8 + Y1^10*X1^8 + 20*Y1^8*Y2^2*X1^6*X2^2 + 60*Y1^6*Y2^4*X1^4*X2^4 + 40*Y1^4*Y2^6*X1^2*X2^6 + 5*Y1^2*Y2^8*X2^8 - 4*Y1^10*Y2*X1^6*X2^2 - 30*Y1^8*Y2^3*X1^4*X2^4 - 40*Y1^6*Y2^5*X1^2*X2^6 - 10*Y1^4*Y2^7*X2^8 + 6*Y1^10*Y2^2*X1^4*X2^4 + 20*Y1^8*Y2^4*X1^2*X2^6 + 10*Y1^6*Y2^6*X2^8 - 4*Y1^10*Y2^3*X1^2*X2^6 - 5*Y1^8*Y2^5*X2^8 + Y1^10*Y2^4*X2^8 ψ₂(x⁴y⁵): p₁ = Y1^5*X2^2 + 6*Y1^5*X1^2*X2 + Y1^5*X1^4 + 20*Y1^4*Y2*X1*X2^2 + 20*Y1^4*Y2*X1^3*X2 + 10*Y1^3*Y2^2*X2^3 + 60*Y1^3*Y2^2*X1^2*X2^2 + 10*Y1^3*Y2^2*X1^4*X2 + 40*Y1^2*Y2^3*X1*X2^3 + 40*Y1^2*Y2^3*X1^3*X2^2 + 5*Y1*Y2^4*X2^4 + 30*Y1*Y2^4*X1^2*X2^3 + 5*Y1*Y2^4*X1^4*X2^2 + 4*Y2^5*X1*X2^4 + 4*Y2^5*X1^3*X2^3 p₂ = Y1^10*X2^4 - 4*Y1^10*X1^2*X2^3 + 6*Y1^10*X1^4*X2^2 - 4*Y1^10*X1^6*X2 + Y1^10*X1^8 - 5*Y1^8*Y2^2*X2^5 + 20*Y1^8*Y2^2*X1^2*X2^4 - 30*Y1^8*Y2^2*X1^4*X2^3 + 20*Y1^8*Y2^2*X1^6*X2^2 - 5*Y1^8*Y2^2*X1^8*X2 + 10*Y1^6*Y2^4*X2^6 - 40*Y1^6*Y2^4*X1^2*X2^5 + 60*Y1^6*Y2^4*X1^4*X2^4 - 40*Y1^6*Y2^4*X1^6*X2^3 + 10*Y1^6*Y2^4*X1^8*X2^2 - 10*Y1^4*Y2^6*X2^7 + 40*Y1^4*Y2^6*X1^2*X2^6 - 60*Y1^4*Y2^6*X1^4*X2^5 + 40*Y1^4*Y2^6*X1^6*X2^4 - 10*Y1^4*Y2^6*X1^8*X2^3 + 5*Y1^2*Y2^8*X2^8 - 20*Y1^2*Y2^8*X1^2*X2^7 + 30*Y1^2*Y2^8*X1^4*X2^6 - 20*Y1^2*Y2^8*X1^6*X2^5 + 5*Y1^2*Y2^8*X1^8*X2^4 - Y2^10*X2^9 + 4*Y2^10*X1^2*X2^8 - 6*Y2^10*X1^4*X2^7 + 4*Y2^10*X1^6*X2^6 - Y2^10*X1^8*X2^5 --------------
Non-homogeneous case¶
General strategy¶
We first treat the more complicated atlas $\psi_1$, where the equations are not homogeneous in $Y_i$'s. Recall that it amounts to computing the torsion-free parts of $I^k/(I^k\cap J + I^{k+1})$, where $I=(Y_1,Y_2)$, and $J = (p_1,p_2)$. Let us denote $\mathbf{R} := \mathbb{C}[X_1,X_2]$. We proceed as follows:
- First, we intersect $I^k$ with $J$ using macaulay2
- Next, we truncate the terms of degree $\geq k+1$ in sage. This gives us a collection $C_k$ of elements in $G_k:=I^k/I^{k+1}$
- Back in macaulay2, we consider the $\mathbf{R}$-submodule $M_k\subset G_k$ spanned by $C_k$, and compute the torsion-free quotient of $G_k/M_k$ as a matrix. Since we know the weights of all variables, we can read off the equivariant multiplicity from it
- We keep increasing $k$. Once the whole module is torsion, we terminate.
We also always start from $k=1$, since the zero case is trivial. Let us recall that in the non-homogeneous case, we have $\mathrm{wt}(Y_1) = w$, $\mathrm{wt}(Y_2) = 2w$.
Case $xy^2$¶
Since the usual multiplicity is $4$, we only need to go up to $k=3$.
In [45]:
macaulay2.needsPackage('"Divisor"')
R = macaulay2('QQ[Y1,Y2,X1,X2,MonomialOrder=>ProductOrder{2,2}]','R')
I = macaulay2('ideal{Y1,Y2}')
P1 = macaulay2('Y2*X1 + Y1^2*X1 + 2*Y1*Y2*X2')
P2 = macaulay2('Y2^2*X1^2 - 2*Y1^2*Y2*X1^2 - Y2^3*X2^2 + Y1^4*X1^2 + 2*Y1^2*Y2^2*X2^2 - Y1^4*Y2*X2^2')
J12 = macaulay2.ideal(P1,P2)
print("I∩J:",str(macaulay2.intersect(I,J12).toString())[6:-1])
print("I²∩J:",str(macaulay2.intersect(I^2,J12).toString())[6:-1])
print("I³∩J:",str(macaulay2.intersect(I^3,J12).toString())[6:-1])
I∩J: Y1^2*X1+2*Y1*Y2*X2+Y2*X1,Y1^4*Y2*X2^2-6*Y1^2*Y2^2*X2^2-8*Y1*Y2^2*X1*X2+Y2^3*X2^2-4*Y2^2*X1^2,Y2^4*X2^4-2*Y2^3*X1^2*X2^2+Y2^2*X1^4,Y1^2*Y2^3*X2^3+4*Y1*Y2^3*X1*X2^2+Y2^4*X2^3-2*Y1*Y2^2*X1^3 I²∩J: Y1^2*Y2*X1+2*Y1*Y2^2*X2+Y2^2*X1,Y1^3*X1+2*Y1^2*Y2*X2+Y1*Y2*X1,Y1^4*Y2*X2^2-6*Y1^2*Y2^2*X2^2-8*Y1*Y2^2*X1*X2+Y2^3*X2^2-4*Y2^2*X1^2,Y2^4*X2^4-2*Y2^3*X1^2*X2^2+Y2^2*X1^4,Y1^2*Y2^3*X2^3+4*Y1*Y2^3*X1*X2^2+Y2^4*X2^3-2*Y1*Y2^2*X1^3 I³∩J: Y1^2*Y2^2*X1+2*Y1*Y2^3*X2+Y2^3*X1,Y1^3*Y2*X1+2*Y1^2*Y2^2*X2+Y1*Y2^2*X1,Y1^4*X1+2*Y1^3*Y2*X2+Y1^2*Y2*X1,Y1^4*Y2*X2^2-6*Y1^2*Y2^2*X2^2+4*Y1^2*Y2*X1^2+Y2^3*X2^2,Y2^4*X2^4-Y1^2*Y2*X1^4-2*Y1*Y2^2*X1^3*X2-2*Y2^3*X1^2*X2^2,Y1^2*Y2^3*X2^3+4*Y1*Y2^3*X1*X2^2+Y2^4*X2^3-2*Y1*Y2^2*X1^3
In [46]:
AAA.<X1,X2>=QQ[]
BBB.<Y1,Y2>=AAA[] #redefine for easier access to the coefficients
def toMacau(arr1,k): #parses list of generators into macaulay2-compatible matrix of coefficients
str1 = "matrix{{"
for i in range(k+1):
for P in arr1:
str1 += str(BBB(P)[k-i,i]) + ","
str1 = str1[:-1]
str1 += "},{"
str1 = str1[:-2]
str1 += "}"
return str1
arr1 = [Y1^2*X1+2*Y1*Y2*X2+Y2*X1,Y1^4*Y2*X2^2-6*Y1^2*Y2^2*X2^2-8*Y1*Y2^2*X1*X2+Y2^3*X2^2-4*Y2^2*X1^2,Y2^4*X2^4-2*Y2^3*X1^2*X2^2+Y2^2*X1^4,Y1^2*Y2^3*X2^3+4*Y1*Y2^3*X1*X2^2+Y2^4*X2^3-2*Y1*Y2^2*X1^3]
arr2 = [Y1^2*Y2*X1+2*Y1*Y2^2*X2+Y2^2*X1,Y1^3*X1+2*Y1^2*Y2*X2+Y1*Y2*X1,Y1^4*Y2*X2^2-6*Y1^2*Y2^2*X2^2-8*Y1*Y2^2*X1*X2+Y2^3*X2^2-4*Y2^2*X1^2,Y2^4*X2^4-2*Y2^3*X1^2*X2^2+Y2^2*X1^4,Y1^2*Y2^3*X2^3+4*Y1*Y2^3*X1*X2^2+Y2^4*X2^3-2*Y1*Y2^2*X1^3]
arr3 = [Y1^2*Y2^2*X1+2*Y1*Y2^3*X2+Y2^3*X1,Y1^3*Y2*X1+2*Y1^2*Y2^2*X2+Y1*Y2^2*X1,Y1^4*X1+2*Y1^3*Y2*X2+Y1^2*Y2*X1,Y1^4*Y2*X2^2-6*Y1^2*Y2^2*X2^2+4*Y1^2*Y2*X1^2+Y2^3*X2^2,Y2^4*X2^4-Y1^2*Y2*X1^4-2*Y1*Y2^2*X1^3*X2-2*Y2^3*X1^2*X2^2,Y1^2*Y2^3*X2^3+4*Y1*Y2^3*X1*X2^2+Y2^4*X2^3-2*Y1*Y2^2*X1^3]
#draw gr^i in the shape of a matrix.
Rp = macaulay2('QQ[X1,X2]')
mat = macaulay2(toMacau(arr1,1))
MMM = mat.coker()
print("gr¹:")
print(MMM)
mat = macaulay2(toMacau(arr2,2))
MMM = mat.coker()
print("gr²:")
print(MMM)
mat = macaulay2(toMacau(arr3,3))
MMM = mat.coker()
print("gr³:")
print(MMM)
gr¹:
cokernel | 0 0 0 0 |
| X1 0 0 0 |
gr²:
cokernel | 0 0 0 0 0 |
| 0 X1 0 0 0 |
| X1 0 -4X1^2 X1^4 0 |
gr³:
cokernel | 0 0 0 0 0 0 |
| 0 0 X1 4X1^2 -X1^4 0 |
| 0 X1 0 0 -2X1^3X2 -2X1^3 |
| X1 0 0 X2^2 -2X1^2X2^2 0 |
In matrices above, each column corresponds to a basis element above, and each row to $Y_1^k$, $Y_1^{k-1}Y_2$, … in this order. From these matrices it is clear that $\mathrm{gr}^1_{\mathrm{tf}} = \mathbf{R} Y_1$, $\mathrm{gr}^2_{\mathrm{tf}} = \mathbf{R} Y_1^2$, $\mathrm{gr}^3_{\mathrm{tf}} = \mathbf{R} Y_1^3$. Recalling that $\mathrm{wt}(Y_1) = w$, we have $$m(t) = [4]_{t^w}$$
Case $x^2y^3$¶
Recall that the usual multiplicity is $9$. Since the polynomials $p_1$, $p_2$ have no linear terms, we may start from $\mathrm{gr}^2$.
In [47]:
R = macaulay2('QQ[Y1,Y2,X1,X2,MonomialOrder=>ProductOrder{2,2}]','R')
I = macaulay2('ideal{Y1,Y2}')
P1 = macaulay2('3*Y1*Y2*X1^2 + 2*Y2^2*X1*X2 + Y1^3*X1^2 + 6*Y1^2*Y2*X1*X2 + 3*Y1*Y2^2*X2^2 + Y1^3*Y2*X2^2')
P2 = macaulay2('-Y2^3*X1^4 + 3*Y1^2*Y2^2*X1^4 + 2*Y2^4*X1^2*X2^2 - 3*Y1^4*Y2*X1^4 - 6*Y1^2*Y2^3*X1^2*X2^2 - Y2^5*X2^4 + Y1^6*X1^4 + 6*Y1^4*Y2^2*X1^2*X2^2 + 3*Y1^2*Y2^4*X2^4 - 2*Y1^6*Y2*X1^2*X2^2 - 3*Y1^4*Y2^3*X2^4 + Y1^6*Y2^2*X2^4')
J23 = macaulay2.ideal(P1,P2)
print("I∩J:",str(macaulay2.intersect(I,J23).toString())[6:-1],"\n")
print("I²∩J:",str(macaulay2.intersect(I^2,J23).toString())[6:-1],"\n")
print("I³∩J:",str(macaulay2.intersect(I^3,J23).toString())[6:-1],"\n")
print("I⁴∩J:",str(macaulay2.intersect(I^4,J23).toString())[6:-1],"\n")
print("I⁵∩J:",str(macaulay2.intersect(I^5,J23).toString())[6:-1],"\n")
print("I⁶∩J:",str(macaulay2.intersect(I^6,J23).toString())[6:-1],"\n")
#print("I⁷∩J:",str(macaulay2.intersect(I^7,J23).toString())[6:-1],"\n")
I∩J: Y1^3*Y2*X2^2+Y1^3*X1^2+6*Y1^2*Y2*X1*X2+3*Y1*Y2^2*X2^2+3*Y1*Y2*X1^2+2*Y2^2*X1*X2,4*Y1^6*X1^4+24*Y1^5*Y2*X1^3*X2+21*Y1^2*Y2^4*X2^4-48*Y1^4*Y2*X1^4-352*Y1^3*Y2^2*X1^3*X2-450*Y1^2*Y2^3*X1^2*X2^2-144*Y1*Y2^4*X1*X2^3-Y2^5*X2^4-159*Y1^2*Y2^2*X1^4-264*Y1*Y2^3*X1^3*X2-102*Y2^4*X1^2*X2^2-Y2^3*X1^4,9*Y1^2*Y2^5*X1*X2^5+4*Y1*Y2^6*X2^6-4*Y1^4*Y2^2*X1^5*X2-54*Y1^2*Y2^4*X1^3*X2^3-20*Y1*Y2^5*X1^2*X2^4+3*Y2^6*X1*X2^5+24*Y1^3*Y2^2*X1^6+141*Y1^2*Y2^3*X1^5*X2+44*Y1*Y2^4*X1^4*X2^2-18*Y2^5*X1^3*X2^3+76*Y1*Y2^3*X1^6+51*Y2^4*X1^5*X2,6*Y1^5*Y2^2*X1^5*X2+81*Y1^2*Y2^5*X1^2*X2^4+36*Y1*Y2^6*X1*X2^5+2*Y2^7*X2^6-36*Y1^4*Y2^2*X1^6-290*Y1^3*Y2^3*X1^5*X2-567*Y1^2*Y2^4*X1^4*X2^2-216*Y1*Y2^5*X1^3*X2^3+17*Y2^6*X1^2*X2^4-108*Y1^2*Y2^3*X1^6-312*Y1*Y2^4*X1^5*X2-167*Y2^5*X1^4*X2^2+2*Y2^4*X1^6 I²∩J: Y1^3*Y2*X2^2+Y1^3*X1^2+6*Y1^2*Y2*X1*X2+3*Y1*Y2^2*X2^2+3*Y1*Y2*X1^2+2*Y2^2*X1*X2,4*Y1^6*X1^4+24*Y1^5*Y2*X1^3*X2+21*Y1^2*Y2^4*X2^4-48*Y1^4*Y2*X1^4-352*Y1^3*Y2^2*X1^3*X2-450*Y1^2*Y2^3*X1^2*X2^2-144*Y1*Y2^4*X1*X2^3-Y2^5*X2^4-159*Y1^2*Y2^2*X1^4-264*Y1*Y2^3*X1^3*X2-102*Y2^4*X1^2*X2^2-Y2^3*X1^4,9*Y1^2*Y2^5*X1*X2^5+4*Y1*Y2^6*X2^6-4*Y1^4*Y2^2*X1^5*X2-54*Y1^2*Y2^4*X1^3*X2^3-20*Y1*Y2^5*X1^2*X2^4+3*Y2^6*X1*X2^5+24*Y1^3*Y2^2*X1^6+141*Y1^2*Y2^3*X1^5*X2+44*Y1*Y2^4*X1^4*X2^2-18*Y2^5*X1^3*X2^3+76*Y1*Y2^3*X1^6+51*Y2^4*X1^5*X2,6*Y1^5*Y2^2*X1^5*X2+81*Y1^2*Y2^5*X1^2*X2^4+36*Y1*Y2^6*X1*X2^5+2*Y2^7*X2^6-36*Y1^4*Y2^2*X1^6-290*Y1^3*Y2^3*X1^5*X2-567*Y1^2*Y2^4*X1^4*X2^2-216*Y1*Y2^5*X1^3*X2^3+17*Y2^6*X1^2*X2^4-108*Y1^2*Y2^3*X1^6-312*Y1*Y2^4*X1^5*X2-167*Y2^5*X1^4*X2^2+2*Y2^4*X1^6 I³∩J: Y1^3*Y2^2*X2^2+Y1^3*Y2*X1^2+6*Y1^2*Y2^2*X1*X2+3*Y1*Y2^3*X2^2+3*Y1*Y2^2*X1^2+2*Y2^3*X1*X2,Y1^4*Y2*X2^2+Y1^4*X1^2+6*Y1^3*Y2*X1*X2+3*Y1^2*Y2^2*X2^2+3*Y1^2*Y2*X1^2+2*Y1*Y2^2*X1*X2,4*Y1^6*X1^4+24*Y1^5*Y2*X1^3*X2+21*Y1^2*Y2^4*X2^4-48*Y1^4*Y2*X1^4-352*Y1^3*Y2^2*X1^3*X2-450*Y1^2*Y2^3*X1^2*X2^2-144*Y1*Y2^4*X1*X2^3-Y2^5*X2^4-159*Y1^2*Y2^2*X1^4-264*Y1*Y2^3*X1^3*X2-102*Y2^4*X1^2*X2^2-Y2^3*X1^4,9*Y1^2*Y2^5*X1*X2^5+4*Y1*Y2^6*X2^6-4*Y1^4*Y2^2*X1^5*X2-54*Y1^2*Y2^4*X1^3*X2^3-20*Y1*Y2^5*X1^2*X2^4+3*Y2^6*X1*X2^5+24*Y1^3*Y2^2*X1^6+141*Y1^2*Y2^3*X1^5*X2+44*Y1*Y2^4*X1^4*X2^2-18*Y2^5*X1^3*X2^3+76*Y1*Y2^3*X1^6+51*Y2^4*X1^5*X2,6*Y1^5*Y2^2*X1^5*X2+81*Y1^2*Y2^5*X1^2*X2^4+36*Y1*Y2^6*X1*X2^5+2*Y2^7*X2^6-36*Y1^4*Y2^2*X1^6-290*Y1^3*Y2^3*X1^5*X2-567*Y1^2*Y2^4*X1^4*X2^2-216*Y1*Y2^5*X1^3*X2^3+17*Y2^6*X1^2*X2^4-108*Y1^2*Y2^3*X1^6-312*Y1*Y2^4*X1^5*X2-167*Y2^5*X1^4*X2^2+2*Y2^4*X1^6 I⁴∩J: Y1^3*Y2^3*X2^2+Y1^3*Y2^2*X1^2+6*Y1^2*Y2^3*X1*X2+3*Y1*Y2^4*X2^2+3*Y1*Y2^3*X1^2+2*Y2^4*X1*X2,Y1^4*Y2^2*X2^2+Y1^4*Y2*X1^2+6*Y1^3*Y2^2*X1*X2+3*Y1^2*Y2^3*X2^2+3*Y1^2*Y2^2*X1^2+2*Y1*Y2^3*X1*X2,Y1^5*Y2*X2^2+Y1^5*X1^2+6*Y1^4*Y2*X1*X2+3*Y1^3*Y2^2*X2^2+3*Y1^3*Y2*X1^2+2*Y1^2*Y2^2*X1*X2,4*Y1^6*Y2*X1^4+24*Y1^5*Y2^2*X1^3*X2+21*Y1^2*Y2^5*X2^4-48*Y1^4*Y2^2*X1^4-352*Y1^3*Y2^3*X1^3*X2-450*Y1^2*Y2^4*X1^2*X2^2-144*Y1*Y2^5*X1*X2^3-Y2^6*X2^4-159*Y1^2*Y2^3*X1^4-264*Y1*Y2^4*X1^3*X2-102*Y2^5*X1^2*X2^2-Y2^4*X1^4,2*Y1^7*X1^4+12*Y1^6*Y2*X1^3*X2-24*Y1^5*Y2*X1^4-176*Y1^4*Y2^2*X1^3*X2-135*Y1^2*Y2^4*X1*X2^3-32*Y1*Y2^5*X2^4+156*Y1^3*Y2^2*X1^4+1281*Y1^2*Y2^3*X1^3*X2+624*Y1*Y2^4*X1^2*X2^2-21*Y2^5*X1*X2^3+706*Y1*Y2^3*X1^4+471*Y2^4*X1^3*X2,9*Y1^2*Y2^5*X1*X2^5+4*Y1*Y2^6*X2^6-4*Y1^4*Y2^2*X1^5*X2-54*Y1^2*Y2^4*X1^3*X2^3-20*Y1*Y2^5*X1^2*X2^4+3*Y2^6*X1*X2^5+24*Y1^3*Y2^2*X1^6+141*Y1^2*Y2^3*X1^5*X2+44*Y1*Y2^4*X1^4*X2^2-18*Y2^5*X1^3*X2^3+76*Y1*Y2^3*X1^6+51*Y2^4*X1^5*X2,6*Y1^5*Y2^2*X1^5*X2+81*Y1^2*Y2^5*X1^2*X2^4+36*Y1*Y2^6*X1*X2^5+2*Y2^7*X2^6-36*Y1^4*Y2^2*X1^6-290*Y1^3*Y2^3*X1^5*X2-567*Y1^2*Y2^4*X1^4*X2^2-216*Y1*Y2^5*X1^3*X2^3+17*Y2^6*X1^2*X2^4-108*Y1^2*Y2^3*X1^6-312*Y1*Y2^4*X1^5*X2-167*Y2^5*X1^4*X2^2+2*Y2^4*X1^6 I⁵∩J: Y1^3*Y2^4*X2^2+Y1^3*Y2^3*X1^2+6*Y1^2*Y2^4*X1*X2+3*Y1*Y2^5*X2^2+3*Y1*Y2^4*X1^2+2*Y2^5*X1*X2,Y1^4*Y2^3*X2^2+Y1^4*Y2^2*X1^2+6*Y1^3*Y2^3*X1*X2+3*Y1^2*Y2^4*X2^2+3*Y1^2*Y2^3*X1^2+2*Y1*Y2^4*X1*X2,Y1^5*Y2^2*X2^2+Y1^5*Y2*X1^2+6*Y1^4*Y2^2*X1*X2+3*Y1^3*Y2^3*X2^2+3*Y1^3*Y2^2*X1^2+2*Y1^2*Y2^3*X1*X2,Y1^6*Y2*X2^2+Y1^6*X1^2+6*Y1^5*Y2*X1*X2+3*Y1^4*Y2^2*X2^2+3*Y1^4*Y2*X1^2+2*Y1^3*Y2^2*X1*X2,6*Y1^7*X1^5+40*Y1^6*Y2*X1^4*X2+21*Y1^2*Y2^5*X2^5-96*Y1^5*Y2*X1^5-720*Y1^4*Y2^2*X1^4*X2-1130*Y1^3*Y2^3*X1^3*X2^2-855*Y1^2*Y2^4*X1^2*X2^3-240*Y1*Y2^5*X1*X2^4-Y2^6*X2^5-310*Y1^3*Y2^2*X1^5-600*Y1^2*Y2^3*X1^4*X2-510*Y1*Y2^4*X1^3*X2^2-165*Y2^5*X1^2*X2^3,4*Y1^6*Y2^2*X1^4+24*Y1^5*Y2^3*X1^3*X2+21*Y1^2*Y2^6*X2^4-48*Y1^4*Y2^3*X1^4-352*Y1^3*Y2^4*X1^3*X2-450*Y1^2*Y2^5*X1^2*X2^2-144*Y1*Y2^6*X1*X2^3-Y2^7*X2^4-159*Y1^2*Y2^4*X1^4-264*Y1*Y2^5*X1^3*X2-102*Y2^6*X1^2*X2^2-Y2^5*X1^4,2*Y1^7*Y2*X1^4+12*Y1^6*Y2^2*X1^3*X2-24*Y1^5*Y2^2*X1^4-176*Y1^4*Y2^3*X1^3*X2-135*Y1^2*Y2^5*X1*X2^3-32*Y1*Y2^6*X2^4+156*Y1^3*Y2^3*X1^4+1281*Y1^2*Y2^4*X1^3*X2+624*Y1*Y2^5*X1^2*X2^2-21*Y2^6*X1*X2^3+706*Y1*Y2^4*X1^4+471*Y2^5*X1^3*X2,Y1^8*X1^4+6*Y1^7*Y2*X1^3*X2-12*Y1^6*Y2*X1^4-88*Y1^5*Y2^2*X1^3*X2-16*Y1^2*Y2^5*X2^4+78*Y1^4*Y2^2*X1^4+708*Y1^3*Y2^3*X1^3*X2+717*Y1^2*Y2^4*X1^2*X2^2+192*Y1*Y2^5*X1*X2^3+353*Y1^2*Y2^3*X1^4+438*Y1*Y2^4*X1^3*X2+135*Y2^5*X1^2*X2^2,Y1^6*Y2*X1^5*X2+12*Y1^2*Y2^5*X1*X2^5+3*Y1*Y2^6*X2^6-6*Y1^5*Y2*X1^6-51*Y1^4*Y2^2*X1^5*X2-125*Y1^3*Y2^3*X1^4*X2^2-153*Y1^2*Y2^4*X1^3*X2^3-51*Y1*Y2^5*X1^2*X2^4+2*Y2^6*X1*X2^5-19*Y1^3*Y2^2*X1^6-60*Y1^2*Y2^3*X1^5*X2-90*Y1*Y2^4*X1^4*X2^2-39*Y2^5*X1^3*X2^3,8*Y1^6*Y2*X1^6+54*Y1^5*Y2^2*X1^5*X2+123*Y1^2*Y2^5*X1^2*X2^4+36*Y1*Y2^6*X1*X2^5+2*Y2^7*X2^6-132*Y1^4*Y2^2*X1^6-994*Y1^3*Y2^3*X1^5*X2-1467*Y1^2*Y2^4*X1^4*X2^2-504*Y1*Y2^5*X1^3*X2^3+15*Y2^6*X1^2*X2^4-426*Y1^2*Y2^3*X1^6-840*Y1*Y2^4*X1^5*X2-371*Y2^5*X1^4*X2^2 I⁶∩J: Y1^3*Y2^5*X2^2+Y1^3*Y2^4*X1^2+6*Y1^2*Y2^5*X1*X2+3*Y1*Y2^6*X2^2+3*Y1*Y2^5*X1^2+2*Y2^6*X1*X2,Y1^4*Y2^4*X2^2+Y1^4*Y2^3*X1^2+6*Y1^3*Y2^4*X1*X2+3*Y1^2*Y2^5*X2^2+3*Y1^2*Y2^4*X1^2+2*Y1*Y2^5*X1*X2,Y1^5*Y2^3*X2^2+Y1^5*Y2^2*X1^2+6*Y1^4*Y2^3*X1*X2+3*Y1^3*Y2^4*X2^2+3*Y1^3*Y2^3*X1^2+2*Y1^2*Y2^4*X1*X2,Y1^6*Y2^2*X2^2+Y1^6*Y2*X1^2+6*Y1^5*Y2^2*X1*X2+3*Y1^4*Y2^3*X2^2+3*Y1^4*Y2^2*X1^2+2*Y1^3*Y2^3*X1*X2,Y1^7*Y2*X2^2+Y1^7*X1^2+6*Y1^6*Y2*X1*X2+3*Y1^5*Y2^2*X2^2+3*Y1^5*Y2*X1^2+2*Y1^4*Y2^2*X1*X2,2*Y1^7*X1^6+15*Y1^6*Y2*X1^5*X2+45*Y1^5*Y2^2*X1^4*X2^2+70*Y1^4*Y2^3*X1^3*X2^3+60*Y1^3*Y2^4*X1^2*X2^4+27*Y1^2*Y2^5*X1*X2^5+5*Y1*Y2^6*X2^6+3*Y1^5*Y2*X1^6+15*Y1^4*Y2^2*X1^5*X2+30*Y1^3*Y2^3*X1^4*X2^2+30*Y1^2*Y2^4*X1^3*X2^3+15*Y1*Y2^5*X1^2*X2^4+3*Y2^6*X1*X2^5,6*Y1^7*Y2*X1^5+40*Y1^6*Y2^2*X1^4*X2+21*Y1^2*Y2^6*X2^5-96*Y1^5*Y2^2*X1^5-720*Y1^4*Y2^3*X1^4*X2-1130*Y1^3*Y2^4*X1^3*X2^2-855*Y1^2*Y2^5*X1^2*X2^3-240*Y1*Y2^6*X1*X2^4-Y2^7*X2^5-310*Y1^3*Y2^3*X1^5-600*Y1^2*Y2^4*X1^4*X2-510*Y1*Y2^5*X1^3*X2^2-165*Y2^6*X1^2*X2^3,5*Y1^8*X1^4*X2-12*Y1^7*X1^5-120*Y1^6*Y2*X1^4*X2-220*Y1^5*Y2^2*X1^3*X2^2-195*Y1^4*Y2^3*X1^2*X2^3-90*Y1^3*Y2^4*X1*X2^4-17*Y1^2*Y2^5*X2^5-68*Y1^5*Y2*X1^5-165*Y1^4*Y2^2*X1^4*X2-180*Y1^3*Y2^3*X1^3*X2^2-105*Y1^2*Y2^4*X1^2*X2^3-30*Y1*Y2^5*X1*X2^4-3*Y2^6*X2^5,3*Y1^8*X1^5+20*Y1^7*Y2*X1^4*X2-48*Y1^6*Y2*X1^5-360*Y1^5*Y2^2*X1^4*X2-565*Y1^4*Y2^3*X1^3*X2^2-438*Y1^3*Y2^4*X1^2*X2^3-183*Y1^2*Y2^5*X1*X2^4-32*Y1*Y2^6*X2^5-155*Y1^4*Y2^2*X1^5-300*Y1^3*Y2^3*X1^4*X2-255*Y1^2*Y2^4*X1^3*X2^2-114*Y1*Y2^5*X1^2*X2^3-21*Y2^6*X1*X2^4,4*Y1^6*Y2^3*X1^4+24*Y1^5*Y2^4*X1^3*X2+21*Y1^2*Y2^7*X2^4-48*Y1^4*Y2^4*X1^4-352*Y1^3*Y2^5*X1^3*X2-450*Y1^2*Y2^6*X1^2*X2^2-144*Y1*Y2^7*X1*X2^3-Y2^8*X2^4-159*Y1^2*Y2^5*X1^4-264*Y1*Y2^6*X1^3*X2-102*Y2^7*X1^2*X2^2-Y2^6*X1^4,2*Y1^7*Y2^2*X1^4+12*Y1^6*Y2^3*X1^3*X2-24*Y1^5*Y2^3*X1^4-176*Y1^4*Y2^4*X1^3*X2-135*Y1^2*Y2^6*X1*X2^3-32*Y1*Y2^7*X2^4+156*Y1^3*Y2^4*X1^4+1281*Y1^2*Y2^5*X1^3*X2+624*Y1*Y2^6*X1^2*X2^2-21*Y2^7*X1*X2^3+706*Y1*Y2^5*X1^4+471*Y2^6*X1^3*X2,Y1^8*Y2*X1^4+6*Y1^7*Y2^2*X1^3*X2-12*Y1^6*Y2^2*X1^4-88*Y1^5*Y2^3*X1^3*X2-16*Y1^2*Y2^6*X2^4+78*Y1^4*Y2^3*X1^4+708*Y1^3*Y2^4*X1^3*X2+717*Y1^2*Y2^5*X1^2*X2^2+192*Y1*Y2^6*X1*X2^3+353*Y1^2*Y2^4*X1^4+438*Y1*Y2^5*X1^3*X2+135*Y2^6*X1^2*X2^2,Y1^9*X1^4+6*Y1^8*Y2*X1^3*X2-12*Y1^7*Y2*X1^4-88*Y1^6*Y2^2*X1^3*X2+78*Y1^5*Y2^2*X1^4+708*Y1^4*Y2^3*X1^3*X2+733*Y1^3*Y2^4*X1^2*X2^2+288*Y1^2*Y2^5*X1*X2^3+48*Y1*Y2^6*X2^4+353*Y1^3*Y2^3*X1^4+438*Y1^2*Y2^4*X1^3*X2+183*Y1*Y2^5*X1^2*X2^2+32*Y2^6*X1*X2^3,Y1^6*Y2*X1^6*X2+3*Y1^5*Y2^2*X1^5*X2^2-10*Y1^3*Y2^4*X1^3*X2^4-15*Y1^2*Y2^5*X1^2*X2^5-9*Y1*Y2^6*X1*X2^6-2*Y2^7*X2^7+3*Y1^5*Y2*X1^7+15*Y1^4*Y2^2*X1^6*X2+30*Y1^3*Y2^3*X1^5*X2^2+30*Y1^2*Y2^4*X1^4*X2^3+15*Y1*Y2^5*X1^3*X2^4+3*Y2^6*X1^2*X2^5,8*Y1^6*Y2^2*X1^6+54*Y1^5*Y2^3*X1^5*X2+123*Y1^2*Y2^6*X1^2*X2^4+36*Y1*Y2^7*X1*X2^5+2*Y2^8*X2^6-132*Y1^4*Y2^3*X1^6-994*Y1^3*Y2^4*X1^5*X2-1467*Y1^2*Y2^5*X1^4*X2^2-504*Y1*Y2^6*X1^3*X2^3+15*Y2^7*X1^2*X2^4-426*Y1^2*Y2^4*X1^6-840*Y1*Y2^5*X1^5*X2-371*Y2^6*X1^4*X2^2
In [48]:
arr2 = [Y1^3*Y2*X2^2+Y1^3*X1^2+6*Y1^2*Y2*X1*X2+3*Y1*Y2^2*X2^2+3*Y1*Y2*X1^2+2*Y2^2*X1*X2,4*Y1^6*X1^4+24*Y1^5*Y2*X1^3*X2+21*Y1^2*Y2^4*X2^4-48*Y1^4*Y2*X1^4-352*Y1^3*Y2^2*X1^3*X2-450*Y1^2*Y2^3*X1^2*X2^2-144*Y1*Y2^4*X1*X2^3-Y2^5*X2^4-159*Y1^2*Y2^2*X1^4-264*Y1*Y2^3*X1^3*X2-102*Y2^4*X1^2*X2^2-Y2^3*X1^4,9*Y1^2*Y2^5*X1*X2^5+4*Y1*Y2^6*X2^6-4*Y1^4*Y2^2*X1^5*X2-54*Y1^2*Y2^4*X1^3*X2^3-20*Y1*Y2^5*X1^2*X2^4+3*Y2^6*X1*X2^5+24*Y1^3*Y2^2*X1^6+141*Y1^2*Y2^3*X1^5*X2+44*Y1*Y2^4*X1^4*X2^2-18*Y2^5*X1^3*X2^3+76*Y1*Y2^3*X1^6+51*Y2^4*X1^5*X2,6*Y1^5*Y2^2*X1^5*X2+81*Y1^2*Y2^5*X1^2*X2^4+36*Y1*Y2^6*X1*X2^5+2*Y2^7*X2^6-36*Y1^4*Y2^2*X1^6-290*Y1^3*Y2^3*X1^5*X2-567*Y1^2*Y2^4*X1^4*X2^2-216*Y1*Y2^5*X1^3*X2^3+17*Y2^6*X1^2*X2^4-108*Y1^2*Y2^3*X1^6-312*Y1*Y2^4*X1^5*X2-167*Y2^5*X1^4*X2^2+2*Y2^4*X1^6]
arr3 = [Y1^3*Y2^2*X2^2+Y1^3*Y2*X1^2+6*Y1^2*Y2^2*X1*X2+3*Y1*Y2^3*X2^2+3*Y1*Y2^2*X1^2+2*Y2^3*X1*X2,Y1^4*Y2*X2^2+Y1^4*X1^2+6*Y1^3*Y2*X1*X2+3*Y1^2*Y2^2*X2^2+3*Y1^2*Y2*X1^2+2*Y1*Y2^2*X1*X2,4*Y1^6*X1^4+24*Y1^5*Y2*X1^3*X2+21*Y1^2*Y2^4*X2^4-48*Y1^4*Y2*X1^4-352*Y1^3*Y2^2*X1^3*X2-450*Y1^2*Y2^3*X1^2*X2^2-144*Y1*Y2^4*X1*X2^3-Y2^5*X2^4-159*Y1^2*Y2^2*X1^4-264*Y1*Y2^3*X1^3*X2-102*Y2^4*X1^2*X2^2-Y2^3*X1^4,9*Y1^2*Y2^5*X1*X2^5+4*Y1*Y2^6*X2^6-4*Y1^4*Y2^2*X1^5*X2-54*Y1^2*Y2^4*X1^3*X2^3-20*Y1*Y2^5*X1^2*X2^4+3*Y2^6*X1*X2^5+24*Y1^3*Y2^2*X1^6+141*Y1^2*Y2^3*X1^5*X2+44*Y1*Y2^4*X1^4*X2^2-18*Y2^5*X1^3*X2^3+76*Y1*Y2^3*X1^6+51*Y2^4*X1^5*X2,6*Y1^5*Y2^2*X1^5*X2+81*Y1^2*Y2^5*X1^2*X2^4+36*Y1*Y2^6*X1*X2^5+2*Y2^7*X2^6-36*Y1^4*Y2^2*X1^6-290*Y1^3*Y2^3*X1^5*X2-567*Y1^2*Y2^4*X1^4*X2^2-216*Y1*Y2^5*X1^3*X2^3+17*Y2^6*X1^2*X2^4-108*Y1^2*Y2^3*X1^6-312*Y1*Y2^4*X1^5*X2-167*Y2^5*X1^4*X2^2+2*Y2^4*X1^6]
arr4 = [Y1^3*Y2^3*X2^2+Y1^3*Y2^2*X1^2+6*Y1^2*Y2^3*X1*X2+3*Y1*Y2^4*X2^2+3*Y1*Y2^3*X1^2+2*Y2^4*X1*X2,Y1^4*Y2^2*X2^2+Y1^4*Y2*X1^2+6*Y1^3*Y2^2*X1*X2+3*Y1^2*Y2^3*X2^2+3*Y1^2*Y2^2*X1^2+2*Y1*Y2^3*X1*X2,Y1^5*Y2*X2^2+Y1^5*X1^2+6*Y1^4*Y2*X1*X2+3*Y1^3*Y2^2*X2^2+3*Y1^3*Y2*X1^2+2*Y1^2*Y2^2*X1*X2,4*Y1^6*Y2*X1^4+24*Y1^5*Y2^2*X1^3*X2+21*Y1^2*Y2^5*X2^4-48*Y1^4*Y2^2*X1^4-352*Y1^3*Y2^3*X1^3*X2-450*Y1^2*Y2^4*X1^2*X2^2-144*Y1*Y2^5*X1*X2^3-Y2^6*X2^4-159*Y1^2*Y2^3*X1^4-264*Y1*Y2^4*X1^3*X2-102*Y2^5*X1^2*X2^2-Y2^4*X1^4,2*Y1^7*X1^4+12*Y1^6*Y2*X1^3*X2-24*Y1^5*Y2*X1^4-176*Y1^4*Y2^2*X1^3*X2-135*Y1^2*Y2^4*X1*X2^3-32*Y1*Y2^5*X2^4+156*Y1^3*Y2^2*X1^4+1281*Y1^2*Y2^3*X1^3*X2+624*Y1*Y2^4*X1^2*X2^2-21*Y2^5*X1*X2^3+706*Y1*Y2^3*X1^4+471*Y2^4*X1^3*X2,9*Y1^2*Y2^5*X1*X2^5+4*Y1*Y2^6*X2^6-4*Y1^4*Y2^2*X1^5*X2-54*Y1^2*Y2^4*X1^3*X2^3-20*Y1*Y2^5*X1^2*X2^4+3*Y2^6*X1*X2^5+24*Y1^3*Y2^2*X1^6+141*Y1^2*Y2^3*X1^5*X2+44*Y1*Y2^4*X1^4*X2^2-18*Y2^5*X1^3*X2^3+76*Y1*Y2^3*X1^6+51*Y2^4*X1^5*X2,6*Y1^5*Y2^2*X1^5*X2+81*Y1^2*Y2^5*X1^2*X2^4+36*Y1*Y2^6*X1*X2^5+2*Y2^7*X2^6-36*Y1^4*Y2^2*X1^6-290*Y1^3*Y2^3*X1^5*X2-567*Y1^2*Y2^4*X1^4*X2^2-216*Y1*Y2^5*X1^3*X2^3+17*Y2^6*X1^2*X2^4-108*Y1^2*Y2^3*X1^6-312*Y1*Y2^4*X1^5*X2-167*Y2^5*X1^4*X2^2+2*Y2^4*X1^6]
arr5 = [Y1^3*Y2^4*X2^2+Y1^3*Y2^3*X1^2+6*Y1^2*Y2^4*X1*X2+3*Y1*Y2^5*X2^2+3*Y1*Y2^4*X1^2+2*Y2^5*X1*X2,Y1^4*Y2^3*X2^2+Y1^4*Y2^2*X1^2+6*Y1^3*Y2^3*X1*X2+3*Y1^2*Y2^4*X2^2+3*Y1^2*Y2^3*X1^2+2*Y1*Y2^4*X1*X2,Y1^5*Y2^2*X2^2+Y1^5*Y2*X1^2+6*Y1^4*Y2^2*X1*X2+3*Y1^3*Y2^3*X2^2+3*Y1^3*Y2^2*X1^2+2*Y1^2*Y2^3*X1*X2,Y1^6*Y2*X2^2+Y1^6*X1^2+6*Y1^5*Y2*X1*X2+3*Y1^4*Y2^2*X2^2+3*Y1^4*Y2*X1^2+2*Y1^3*Y2^2*X1*X2,6*Y1^7*X1^5+40*Y1^6*Y2*X1^4*X2+21*Y1^2*Y2^5*X2^5-96*Y1^5*Y2*X1^5-720*Y1^4*Y2^2*X1^4*X2-1130*Y1^3*Y2^3*X1^3*X2^2-855*Y1^2*Y2^4*X1^2*X2^3-240*Y1*Y2^5*X1*X2^4-Y2^6*X2^5-310*Y1^3*Y2^2*X1^5-600*Y1^2*Y2^3*X1^4*X2-510*Y1*Y2^4*X1^3*X2^2-165*Y2^5*X1^2*X2^3,4*Y1^6*Y2^2*X1^4+24*Y1^5*Y2^3*X1^3*X2+21*Y1^2*Y2^6*X2^4-48*Y1^4*Y2^3*X1^4-352*Y1^3*Y2^4*X1^3*X2-450*Y1^2*Y2^5*X1^2*X2^2-144*Y1*Y2^6*X1*X2^3-Y2^7*X2^4-159*Y1^2*Y2^4*X1^4-264*Y1*Y2^5*X1^3*X2-102*Y2^6*X1^2*X2^2-Y2^5*X1^4,2*Y1^7*Y2*X1^4+12*Y1^6*Y2^2*X1^3*X2-24*Y1^5*Y2^2*X1^4-176*Y1^4*Y2^3*X1^3*X2-135*Y1^2*Y2^5*X1*X2^3-32*Y1*Y2^6*X2^4+156*Y1^3*Y2^3*X1^4+1281*Y1^2*Y2^4*X1^3*X2+624*Y1*Y2^5*X1^2*X2^2-21*Y2^6*X1*X2^3+706*Y1*Y2^4*X1^4+471*Y2^5*X1^3*X2,Y1^8*X1^4+6*Y1^7*Y2*X1^3*X2-12*Y1^6*Y2*X1^4-88*Y1^5*Y2^2*X1^3*X2-16*Y1^2*Y2^5*X2^4+78*Y1^4*Y2^2*X1^4+708*Y1^3*Y2^3*X1^3*X2+717*Y1^2*Y2^4*X1^2*X2^2+192*Y1*Y2^5*X1*X2^3+353*Y1^2*Y2^3*X1^4+438*Y1*Y2^4*X1^3*X2+135*Y2^5*X1^2*X2^2,Y1^6*Y2*X1^5*X2+12*Y1^2*Y2^5*X1*X2^5+3*Y1*Y2^6*X2^6-6*Y1^5*Y2*X1^6-51*Y1^4*Y2^2*X1^5*X2-125*Y1^3*Y2^3*X1^4*X2^2-153*Y1^2*Y2^4*X1^3*X2^3-51*Y1*Y2^5*X1^2*X2^4+2*Y2^6*X1*X2^5-19*Y1^3*Y2^2*X1^6-60*Y1^2*Y2^3*X1^5*X2-90*Y1*Y2^4*X1^4*X2^2-39*Y2^5*X1^3*X2^3,8*Y1^6*Y2*X1^6+54*Y1^5*Y2^2*X1^5*X2+123*Y1^2*Y2^5*X1^2*X2^4+36*Y1*Y2^6*X1*X2^5+2*Y2^7*X2^6-132*Y1^4*Y2^2*X1^6-994*Y1^3*Y2^3*X1^5*X2-1467*Y1^2*Y2^4*X1^4*X2^2-504*Y1*Y2^5*X1^3*X2^3+15*Y2^6*X1^2*X2^4-426*Y1^2*Y2^3*X1^6-840*Y1*Y2^4*X1^5*X2-371*Y2^5*X1^4*X2^2]
arr6 = [Y1^3*Y2^5*X2^2+Y1^3*Y2^4*X1^2+6*Y1^2*Y2^5*X1*X2+3*Y1*Y2^6*X2^2+3*Y1*Y2^5*X1^2+2*Y2^6*X1*X2,Y1^4*Y2^4*X2^2+Y1^4*Y2^3*X1^2+6*Y1^3*Y2^4*X1*X2+3*Y1^2*Y2^5*X2^2+3*Y1^2*Y2^4*X1^2+2*Y1*Y2^5*X1*X2,Y1^5*Y2^3*X2^2+Y1^5*Y2^2*X1^2+6*Y1^4*Y2^3*X1*X2+3*Y1^3*Y2^4*X2^2+3*Y1^3*Y2^3*X1^2+2*Y1^2*Y2^4*X1*X2,Y1^6*Y2^2*X2^2+Y1^6*Y2*X1^2+6*Y1^5*Y2^2*X1*X2+3*Y1^4*Y2^3*X2^2+3*Y1^4*Y2^2*X1^2+2*Y1^3*Y2^3*X1*X2,Y1^7*Y2*X2^2+Y1^7*X1^2+6*Y1^6*Y2*X1*X2+3*Y1^5*Y2^2*X2^2+3*Y1^5*Y2*X1^2+2*Y1^4*Y2^2*X1*X2,2*Y1^7*X1^6+15*Y1^6*Y2*X1^5*X2+45*Y1^5*Y2^2*X1^4*X2^2+70*Y1^4*Y2^3*X1^3*X2^3+60*Y1^3*Y2^4*X1^2*X2^4+27*Y1^2*Y2^5*X1*X2^5+5*Y1*Y2^6*X2^6+3*Y1^5*Y2*X1^6+15*Y1^4*Y2^2*X1^5*X2+30*Y1^3*Y2^3*X1^4*X2^2+30*Y1^2*Y2^4*X1^3*X2^3+15*Y1*Y2^5*X1^2*X2^4+3*Y2^6*X1*X2^5,6*Y1^7*Y2*X1^5+40*Y1^6*Y2^2*X1^4*X2+21*Y1^2*Y2^6*X2^5-96*Y1^5*Y2^2*X1^5-720*Y1^4*Y2^3*X1^4*X2-1130*Y1^3*Y2^4*X1^3*X2^2-855*Y1^2*Y2^5*X1^2*X2^3-240*Y1*Y2^6*X1*X2^4-Y2^7*X2^5-310*Y1^3*Y2^3*X1^5-600*Y1^2*Y2^4*X1^4*X2-510*Y1*Y2^5*X1^3*X2^2-165*Y2^6*X1^2*X2^3,5*Y1^8*X1^4*X2-12*Y1^7*X1^5-120*Y1^6*Y2*X1^4*X2-220*Y1^5*Y2^2*X1^3*X2^2-195*Y1^4*Y2^3*X1^2*X2^3-90*Y1^3*Y2^4*X1*X2^4-17*Y1^2*Y2^5*X2^5-68*Y1^5*Y2*X1^5-165*Y1^4*Y2^2*X1^4*X2-180*Y1^3*Y2^3*X1^3*X2^2-105*Y1^2*Y2^4*X1^2*X2^3-30*Y1*Y2^5*X1*X2^4-3*Y2^6*X2^5,3*Y1^8*X1^5+20*Y1^7*Y2*X1^4*X2-48*Y1^6*Y2*X1^5-360*Y1^5*Y2^2*X1^4*X2-565*Y1^4*Y2^3*X1^3*X2^2-438*Y1^3*Y2^4*X1^2*X2^3-183*Y1^2*Y2^5*X1*X2^4-32*Y1*Y2^6*X2^5-155*Y1^4*Y2^2*X1^5-300*Y1^3*Y2^3*X1^4*X2-255*Y1^2*Y2^4*X1^3*X2^2-114*Y1*Y2^5*X1^2*X2^3-21*Y2^6*X1*X2^4,4*Y1^6*Y2^3*X1^4+24*Y1^5*Y2^4*X1^3*X2+21*Y1^2*Y2^7*X2^4-48*Y1^4*Y2^4*X1^4-352*Y1^3*Y2^5*X1^3*X2-450*Y1^2*Y2^6*X1^2*X2^2-144*Y1*Y2^7*X1*X2^3-Y2^8*X2^4-159*Y1^2*Y2^5*X1^4-264*Y1*Y2^6*X1^3*X2-102*Y2^7*X1^2*X2^2-Y2^6*X1^4,2*Y1^7*Y2^2*X1^4+12*Y1^6*Y2^3*X1^3*X2-24*Y1^5*Y2^3*X1^4-176*Y1^4*Y2^4*X1^3*X2-135*Y1^2*Y2^6*X1*X2^3-32*Y1*Y2^7*X2^4+156*Y1^3*Y2^4*X1^4+1281*Y1^2*Y2^5*X1^3*X2+624*Y1*Y2^6*X1^2*X2^2-21*Y2^7*X1*X2^3+706*Y1*Y2^5*X1^4+471*Y2^6*X1^3*X2,Y1^8*Y2*X1^4+6*Y1^7*Y2^2*X1^3*X2-12*Y1^6*Y2^2*X1^4-88*Y1^5*Y2^3*X1^3*X2-16*Y1^2*Y2^6*X2^4+78*Y1^4*Y2^3*X1^4+708*Y1^3*Y2^4*X1^3*X2+717*Y1^2*Y2^5*X1^2*X2^2+192*Y1*Y2^6*X1*X2^3+353*Y1^2*Y2^4*X1^4+438*Y1*Y2^5*X1^3*X2+135*Y2^6*X1^2*X2^2,Y1^9*X1^4+6*Y1^8*Y2*X1^3*X2-12*Y1^7*Y2*X1^4-88*Y1^6*Y2^2*X1^3*X2+78*Y1^5*Y2^2*X1^4+708*Y1^4*Y2^3*X1^3*X2+733*Y1^3*Y2^4*X1^2*X2^2+288*Y1^2*Y2^5*X1*X2^3+48*Y1*Y2^6*X2^4+353*Y1^3*Y2^3*X1^4+438*Y1^2*Y2^4*X1^3*X2+183*Y1*Y2^5*X1^2*X2^2+32*Y2^6*X1*X2^3,Y1^6*Y2*X1^6*X2+3*Y1^5*Y2^2*X1^5*X2^2-10*Y1^3*Y2^4*X1^3*X2^4-15*Y1^2*Y2^5*X1^2*X2^5-9*Y1*Y2^6*X1*X2^6-2*Y2^7*X2^7+3*Y1^5*Y2*X1^7+15*Y1^4*Y2^2*X1^6*X2+30*Y1^3*Y2^3*X1^5*X2^2+30*Y1^2*Y2^4*X1^4*X2^3+15*Y1*Y2^5*X1^3*X2^4+3*Y2^6*X1^2*X2^5,8*Y1^6*Y2^2*X1^6+54*Y1^5*Y2^3*X1^5*X2+123*Y1^2*Y2^6*X1^2*X2^4+36*Y1*Y2^7*X1*X2^5+2*Y2^8*X2^6-132*Y1^4*Y2^3*X1^6-994*Y1^3*Y2^4*X1^5*X2-1467*Y1^2*Y2^5*X1^4*X2^2-504*Y1*Y2^6*X1^3*X2^3+15*Y2^7*X1^2*X2^4-426*Y1^2*Y2^4*X1^6-840*Y1*Y2^5*X1^5*X2-371*Y2^6*X1^4*X2^2]
#arr7 = [Y1^3*Y2^6*X2^2+Y1^3*Y2^5*X1^2+6*Y1^2*Y2^6*X1*X2+3*Y1*Y2^7*X2^2+3*Y1*Y2^6*X1^2+2*Y2^7*X1*X2,Y1^4*Y2^5*X2^2+Y1^4*Y2^4*X1^2+6*Y1^3*Y2^5*X1*X2+3*Y1^2*Y2^6*X2^2+3*Y1^2*Y2^5*X1^2+2*Y1*Y2^6*X1*X2,Y1^5*Y2^4*X2^2+Y1^5*Y2^3*X1^2+6*Y1^4*Y2^4*X1*X2+3*Y1^3*Y2^5*X2^2+3*Y1^3*Y2^4*X1^2+2*Y1^2*Y2^5*X1*X2,Y1^6*Y2^3*X2^2+Y1^6*Y2^2*X1^2+6*Y1^5*Y2^3*X1*X2+3*Y1^4*Y2^4*X2^2+3*Y1^4*Y2^3*X1^2+2*Y1^3*Y2^4*X1*X2,Y1^7*Y2^2*X2^2+Y1^7*Y2*X1^2+6*Y1^6*Y2^2*X1*X2+3*Y1^5*Y2^3*X2^2+3*Y1^5*Y2^2*X1^2+2*Y1^4*Y2^3*X1*X2,Y1^8*Y2*X2^2+Y1^8*X1^2+6*Y1^7*Y2*X1*X2+3*Y1^6*Y2^2*X2^2+3*Y1^6*Y2*X1^2+2*Y1^5*Y2^2*X1*X2,Y1^7*X1^7+7*Y1^6*Y2*X1^6*X2+21*Y1^5*Y2^2*X1^5*X2^2+35*Y1^4*Y2^3*X1^4*X2^3+35*Y1^3*Y2^4*X1^3*X2^4+21*Y1^2*Y2^5*X1^2*X2^5+7*Y1*Y2^6*X1*X2^6+Y2^7*X2^7,3*Y1^7*Y2*X1^5*X2+15*Y1^6*Y2^2*X1^4*X2^2+30*Y1^5*Y2^3*X1^3*X2^3+30*Y1^4*Y2^4*X1^2*X2^4+15*Y1^3*Y2^5*X1*X2^5+3*Y1^2*Y2^6*X2^6+5*Y1^6*Y2*X1^6+27*Y1^5*Y2^2*X1^5*X2+60*Y1^4*Y2^3*X1^4*X2^2+70*Y1^3*Y2^4*X1^3*X2^3+45*Y1^2*Y2^5*X1^2*X2^4+15*Y1*Y2^6*X1*X2^5+2*Y2^7*X2^6,2*Y1^7*Y2*X1^6+15*Y1^6*Y2^2*X1^5*X2+45*Y1^5*Y2^3*X1^4*X2^2+70*Y1^4*Y2^4*X1^3*X2^3+60*Y1^3*Y2^5*X1^2*X2^4+27*Y1^2*Y2^6*X1*X2^5+5*Y1*Y2^7*X2^6+3*Y1^5*Y2^2*X1^6+15*Y1^4*Y2^3*X1^5*X2+30*Y1^3*Y2^4*X1^4*X2^2+30*Y1^2*Y2^5*X1^3*X2^3+15*Y1*Y2^6*X1^2*X2^4+3*Y2^7*X1*X2^5,3*Y1^8*X1^5*X2+15*Y1^7*Y2*X1^4*X2^2+30*Y1^6*Y2^2*X1^3*X2^3+30*Y1^5*Y2^3*X1^2*X2^4+15*Y1^4*Y2^4*X1*X2^5+3*Y1^3*Y2^5*X2^6+5*Y1^7*X1^6+27*Y1^6*Y2*X1^5*X2+60*Y1^5*Y2^2*X1^4*X2^2+70*Y1^4*Y2^3*X1^3*X2^3+45*Y1^3*Y2^4*X1^2*X2^4+15*Y1^2*Y2^5*X1*X2^5+2*Y1*Y2^6*X2^6,Y1^8*X1^6-15*Y1^6*Y2^2*X1^4*X2^2-40*Y1^5*Y2^3*X1^3*X2^3-45*Y1^4*Y2^4*X1^2*X2^4-24*Y1^3*Y2^5*X1*X2^5-5*Y1^2*Y2^6*X2^6-11*Y1^6*Y2*X1^6-60*Y1^5*Y2^2*X1^5*X2-135*Y1^4*Y2^3*X1^4*X2^2-160*Y1^3*Y2^4*X1^3*X2^3-105*Y1^2*Y2^5*X1^2*X2^4-36*Y1*Y2^6*X1*X2^5-5*Y2^7*X2^6,6*Y1^7*Y2^2*X1^5+40*Y1^6*Y2^3*X1^4*X2+21*Y1^2*Y2^7*X2^5-96*Y1^5*Y2^3*X1^5-720*Y1^4*Y2^4*X1^4*X2-1130*Y1^3*Y2^5*X1^3*X2^2-855*Y1^2*Y2^6*X1^2*X2^3-240*Y1*Y2^7*X1*X2^4-Y2^8*X2^5-310*Y1^3*Y2^4*X1^5-600*Y1^2*Y2^5*X1^4*X2-510*Y1*Y2^6*X1^3*X2^2-165*Y2^7*X1^2*X2^3,5*Y1^8*Y2*X1^4*X2-12*Y1^7*Y2*X1^5-120*Y1^6*Y2^2*X1^4*X2-220*Y1^5*Y2^3*X1^3*X2^2-195*Y1^4*Y2^4*X1^2*X2^3-90*Y1^3*Y2^5*X1*X2^4-17*Y1^2*Y2^6*X2^5-68*Y1^5*Y2^2*X1^5-165*Y1^4*Y2^3*X1^4*X2-180*Y1^3*Y2^4*X1^3*X2^2-105*Y1^2*Y2^5*X1^2*X2^3-30*Y1*Y2^6*X1*X2^4-3*Y2^7*X2^5,3*Y1^8*Y2*X1^5+20*Y1^7*Y2^2*X1^4*X2-48*Y1^6*Y2^2*X1^5-360*Y1^5*Y2^3*X1^4*X2-565*Y1^4*Y2^4*X1^3*X2^2-438*Y1^3*Y2^5*X1^2*X2^3-183*Y1^2*Y2^6*X1*X2^4-32*Y1*Y2^7*X2^5-155*Y1^4*Y2^3*X1^5-300*Y1^3*Y2^4*X1^4*X2-255*Y1^2*Y2^5*X1^3*X2^2-114*Y1*Y2^6*X1^2*X2^3-21*Y2^7*X1*X2^4,5*Y1^9*X1^4*X2-12*Y1^8*X1^5-120*Y1^7*Y2*X1^4*X2-220*Y1^6*Y2^2*X1^3*X2^2-195*Y1^5*Y2^3*X1^2*X2^3-90*Y1^4*Y2^4*X1*X2^4-17*Y1^3*Y2^5*X2^5-68*Y1^6*Y2*X1^5-165*Y1^5*Y2^2*X1^4*X2-180*Y1^4*Y2^3*X1^3*X2^2-105*Y1^3*Y2^4*X1^2*X2^3-30*Y1^2*Y2^5*X1*X2^4-3*Y1*Y2^6*X2^5,Y1^9*X1^5+40*Y1^6*Y2^2*X1^4*X2+105*Y1^5*Y2^3*X1^3*X2^2+114*Y1^4*Y2^4*X1^2*X2^3+59*Y1^3*Y2^5*X1*X2^4+12*Y1^2*Y2^6*X2^5+39*Y1^5*Y2^2*X1^5+120*Y1^4*Y2^3*X1^4*X2+155*Y1^3*Y2^4*X1^3*X2^2+102*Y1^2*Y2^5*X1^2*X2^3+33*Y1*Y2^6*X1*X2^4+4*Y2^7*X2^5,4*Y1^6*Y2^4*X1^4+24*Y1^5*Y2^5*X1^3*X2+21*Y1^2*Y2^8*X2^4-48*Y1^4*Y2^5*X1^4-352*Y1^3*Y2^6*X1^3*X2-450*Y1^2*Y2^7*X1^2*X2^2-144*Y1*Y2^8*X1*X2^3-Y2^9*X2^4-159*Y1^2*Y2^6*X1^4-264*Y1*Y2^7*X1^3*X2-102*Y2^8*X1^2*X2^2-Y2^7*X1^4,2*Y1^7*Y2^3*X1^4+12*Y1^6*Y2^4*X1^3*X2-24*Y1^5*Y2^4*X1^4-176*Y1^4*Y2^5*X1^3*X2-135*Y1^2*Y2^7*X1*X2^3-32*Y1*Y2^8*X2^4+156*Y1^3*Y2^5*X1^4+1281*Y1^2*Y2^6*X1^3*X2+624*Y1*Y2^7*X1^2*X2^2-21*Y2^8*X1*X2^3+706*Y1*Y2^6*X1^4+471*Y2^7*X1^3*X2,Y1^8*Y2^2*X1^4+6*Y1^7*Y2^3*X1^3*X2-12*Y1^6*Y2^3*X1^4-88*Y1^5*Y2^4*X1^3*X2-16*Y1^2*Y2^7*X2^4+78*Y1^4*Y2^4*X1^4+708*Y1^3*Y2^5*X1^3*X2+717*Y1^2*Y2^6*X1^2*X2^2+192*Y1*Y2^7*X1*X2^3+353*Y1^2*Y2^5*X1^4+438*Y1*Y2^6*X1^3*X2+135*Y2^7*X1^2*X2^2,Y1^9*Y2*X1^4+6*Y1^8*Y2^2*X1^3*X2-12*Y1^7*Y2^2*X1^4-88*Y1^6*Y2^3*X1^3*X2+78*Y1^5*Y2^3*X1^4+708*Y1^4*Y2^4*X1^3*X2+733*Y1^3*Y2^5*X1^2*X2^2+288*Y1^2*Y2^6*X1*X2^3+48*Y1*Y2^7*X2^4+353*Y1^3*Y2^4*X1^4+438*Y1^2*Y2^5*X1^3*X2+183*Y1*Y2^6*X1^2*X2^2+32*Y2^7*X1*X2^3,Y1^10*X1^4+6*Y1^9*Y2*X1^3*X2-12*Y1^8*Y2*X1^4-88*Y1^7*Y2^2*X1^3*X2+78*Y1^6*Y2^2*X1^4+708*Y1^5*Y2^3*X1^3*X2+733*Y1^4*Y2^4*X1^2*X2^2+288*Y1^3*Y2^5*X1*X2^3+48*Y1^2*Y2^6*X2^4+353*Y1^4*Y2^3*X1^4+438*Y1^3*Y2^4*X1^3*X2+183*Y1^2*Y2^5*X1^2*X2^2+32*Y1*Y2^6*X1*X2^3,8*Y1^6*Y2^3*X1^6+54*Y1^5*Y2^4*X1^5*X2+123*Y1^2*Y2^7*X1^2*X2^4+36*Y1*Y2^8*X1*X2^5+2*Y2^9*X2^6-132*Y1^4*Y2^4*X1^6-994*Y1^3*Y2^5*X1^5*X2-1467*Y1^2*Y2^6*X1^4*X2^2-504*Y1*Y2^7*X1^3*X2^3+15*Y2^8*X1^2*X2^4-426*Y1^2*Y2^5*X1^6-840*Y1*Y2^6*X1^5*X2-371*Y2^7*X1^4*X2^2]
#draw gr^i in the shape of a matrix.
Rp = macaulay2('QQ[X1,X2]')
mat = macaulay2(toMacau(arr2,2))
MMM2 = mat.coker()
print("gr²:")
print(MMM2)
mat = macaulay2(toMacau(arr3,3))
MMM3 = mat.coker()
print("gr³:")
print(MMM3)
mat = macaulay2(toMacau(arr4,4))
MMM4 = mat.coker()
print("gr⁴:")
print(MMM4)
mat = macaulay2(toMacau(arr5,5))
MMM5 = mat.coker()
print("gr⁵:")
print(MMM5)
mat = macaulay2(toMacau(arr6,6))
MMM6 = mat.coker()
print("gr⁶:")
print(MMM6)
#mat = macaulay2(toMacau(arr7,7))
#MMM7 = mat.coker()
#print("gr⁷:")
#print(MMM7)
gr²:
cokernel | 0 0 0 0 |
| 3X1^2 0 0 0 |
| 2X1X2 0 0 0 |
gr³:
cokernel | 0 0 0 0 0 |
| 0 3X1^2 0 0 0 |
| 3X1^2 2X1X2 0 0 0 |
| 2X1X2 0 -X1^4 0 0 |
gr⁴:
cokernel | 0 0 0 0 0 0 0 |
| 0 0 3X1^2 0 0 0 0 |
| 0 3X1^2 2X1X2 0 0 0 0 |
| 3X1^2 2X1X2 0 0 706X1^4 76X1^6 0 |
| 2X1X2 0 0 -X1^4 471X1^3X2 51X1^5X2 2X1^6 |
gr⁵:
cokernel | 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 3X1^2 0 0 0 0 0 0 |
| 0 0 3X1^2 2X1X2 -310X1^5 0 0 0 -19X1^6 0 |
| 0 3X1^2 2X1X2 0 -600X1^4X2 0 0 353X1^4 -60X1^5X2 -426X1^6 |
| 3X1^2 2X1X2 0 0 -510X1^3X2^2 0 706X1^4 438X1^3X2 -90X1^4X2^2 -840X1^5X2 |
| 2X1X2 0 0 0 -165X1^2X2^3 -X1^4 471X1^3X2 135X1^2X2^2 -39X1^3X2^3 -371X1^4X2^2 |
gr⁶:
cokernel | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 3X1^2 3X1^6 0 -68X1^5 0 0 0 0 0 3X1^7 0 |
| 0 0 0 3X1^2 2X1X2 15X1^5X2 0 -165X1^4X2 -155X1^5 0 0 0 0 15X1^6X2 0 |
| 0 0 3X1^2 2X1X2 0 30X1^4X2^2 -310X1^5 -180X1^3X2^2 -300X1^4X2 0 0 0 353X1^4 30X1^5X2^2 0 |
| 0 3X1^2 2X1X2 0 0 30X1^3X2^3 -600X1^4X2 -105X1^2X2^3 -255X1^3X2^2 0 0 353X1^4 438X1^3X2 30X1^4X2^3 -426X1^6 |
| 3X1^2 2X1X2 0 0 0 15X1^2X2^4 -510X1^3X2^2 -30X1X2^4 -114X1^2X2^3 0 706X1^4 438X1^3X2 183X1^2X2^2 15X1^3X2^4 -840X1^5X2 |
| 2X1X2 0 0 0 0 3X1X2^5 -165X1^2X2^3 -3X2^5 -21X1X2^4 -X1^4 471X1^3X2 135X1^2X2^2 32X1X2^3 3X1^2X2^5 -371X1^4X2^2 |
In [49]:
NN = MMM2 / MMM2.torsionSubmodule()
print("gr²:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM3 / MMM3.torsionSubmodule()
print("gr³:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM4 / MMM4.torsionSubmodule()
print("gr⁴:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM5 / MMM5.torsionSubmodule()
print("gr⁵:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM6 / MMM6.torsionSubmodule()
print("gr⁶:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
#NN = MMM7 / MMM7.torsionSubmodule()
#print(NN)
#print(NN.prune())
#print(NN.rank(),"\n-------------------")
gr²:
cokernel | 0 |
| 3X1 |
| 2X2 |
rank = 2
-------------------
gr³:
1
(QQ[X1, X2])
rank = 1
-------------------
gr⁴:
1
(QQ[X1, X2])
rank = 1
-------------------
gr⁵:
1
(QQ[X1, X2])
rank = 1
-------------------
gr⁶:
1
(QQ[X1, X2])
rank = 1
-------------------
From these matrices it is clear that $\mathrm{gr}^3_{\mathrm{tf}} = \mathbf{R} Y_1^3$, $\mathrm{gr}^4_{\mathrm{tf}} = \mathbf{R} Y_1^4$, $\mathrm{gr}^5_{\mathrm{tf}} = \mathbf{R} Y_1^5$, $\mathrm{gr}^6_{\mathrm{tf}} = \mathbf{R} Y_1^6$. We also clearly have $\mathrm{gr}^1_{\mathrm{tf}} = \mathbf{R} Y_1 \oplus \mathbf{R} Y_2$. For $\mathrm{gr}^2_{\mathrm{tf}}$, we have a sum of $\mathbf{R} Y_1^2$ and a torsion-free module of rank $2$ which is not locally free. Recalling that $\mathrm{wt}(X_1) = -l = -w + 1$, $\mathrm{wt}(X_2) = -l-w = -2w + 1$, we get an equality in $K$-theory $$[\mathrm{gr}^2_{\mathrm{tf}}] = t^{3w}+t^{4w}-t^{2w+1}$$ Recalling that $\mathrm{wt}(Y_1) = w$ and summing everything up, we obtain $$m(t) = [7]_{t^w} + t^{2w} + t^{3w}+t^{4w}-t^{2w+1}$$
Case $x^2y^4$¶
Here, the usual multiplicity is $16$. Since the polynomials $p_1$, $p_2$ have no linear terms, we may start from $\mathrm{gr}^2$.
In [50]:
R = macaulay2('QQ[Y1,Y2,X1,X2,MonomialOrder=>ProductOrder{2,2}]','R')
I = macaulay2('ideal{Y1,Y2}')
P1 = macaulay2('Y2^2*X1^2 + 6*Y1^2*Y2*X1^2 + 8*Y1*Y2^2*X1*X2 + Y2^3*X2^2 + Y1^4*X1^2 + 8*Y1^3*Y2*X1*X2 + 6*Y1^2*Y2^2*X2^2 + Y1^4*Y2*X2^2')
P2 = macaulay2('Y2^4*X1^4 - 4*Y1^2*Y2^3*X1^4 - 2*Y2^5*X1^2*X2^2 + 6*Y1^4*Y2^2*X1^4 + 8*Y1^2*Y2^4*X1^2*X2^2 + Y2^6*X2^4 - 4*Y1^6*Y2*X1^4 - 12*Y1^4*Y2^3*X1^2*X2^2 - 4*Y1^2*Y2^5*X2^4 + Y1^8*X1^4 + 8*Y1^6*Y2^2*X1^2*X2^2 + 6*Y1^4*Y2^4*X2^4 - 2*Y1^8*Y2*X1^2*X2^2 - 4*Y1^6*Y2^3*X2^4 + Y1^8*Y2^2*X2^4')
J24 = macaulay2.ideal(P1,P2)
#print("I∩J:",str(macaulay2.intersect(I,J24).toString())[6:-1],"\n")
print("I²∩J:",str(macaulay2.intersect(I^2,J24).toString())[6:-1],"\n")
print("I³∩J:",str(macaulay2.intersect(I^3,J24).toString())[6:-1],"\n")
print("I⁴∩J:",str(macaulay2.intersect(I^4,J24).toString())[6:-1],"\n")
print("I⁵∩J:",str(macaulay2.intersect(I^5,J24).toString())[6:-1],"\n")
print("I⁶∩J:",str(macaulay2.intersect(I^6,J24).toString())[6:-1],"\n")
print("I⁷∩J:",str(macaulay2.intersect(I^7,J24).toString())[6:-1],"\n")
print("I⁸∩J:",str(macaulay2.intersect(I^8,J24).toString())[6:-1],"\n")
print("I⁹∩J:",str(macaulay2.intersect(I^9,J24).toString())[6:-1],"\n")
I²∩J: Y1^4*Y2*X2^2+Y1^4*X1^2+8*Y1^3*Y2*X1*X2+6*Y1^2*Y2^2*X2^2+6*Y1^2*Y2*X1^2+8*Y1*Y2^2*X1*X2+Y2^3*X2^2+Y2^2*X1^2,Y1^8*X1^4+8*Y1^7*Y2*X1^3*X2-20*Y1^6*Y2*X1^4-200*Y1^5*Y2^2*X1^3*X2-288*Y1^3*Y2^4*X1*X2^3-96*Y1^2*Y2^5*X2^4+230*Y1^4*Y2^2*X1^4+2584*Y1^3*Y2^3*X1^3*X2+1856*Y1^2*Y2^4*X1^2*X2^2-160*Y1*Y2^5*X1*X2^3-16*Y2^6*X2^4+2188*Y1^2*Y2^3*X1^4+2920*Y1*Y2^4*X1^3*X2+352*Y2^5*X1^2*X2^2+369*Y2^4*X1^4,Y1^7*Y2*X1^5*X2+36*Y1^3*Y2^5*X1*X2^5+12*Y1^2*Y2^6*X2^6-8*Y1^6*Y2*X1^6-89*Y1^5*Y2^2*X1^5*X2-472*Y1^3*Y2^4*X1^3*X2^3-220*Y1^2*Y2^5*X1^2*X2^4+20*Y1*Y2^6*X1*X2^5+2*Y2^7*X2^6+240*Y1^4*Y2^2*X1^6+2091*Y1^3*Y2^3*X1^5*X2+1252*Y1^2*Y2^4*X1^4*X2^2-376*Y1*Y2^5*X1^3*X2^3-42*Y2^6*X1^2*X2^4+1732*Y1^2*Y2^3*X1^6+2293*Y1*Y2^4*X1^5*X2+246*Y2^5*X1^4*X2^2+290*Y2^4*X1^6,5*Y1^7*Y2*X1^7+45*Y1^6*Y2^2*X1^6*X2+244*Y1^3*Y2^5*X1^3*X2^4+100*Y1^2*Y2^6*X1^2*X2^5+10*Y1*Y2^7*X1*X2^6+2*Y2^8*X2^7-141*Y1^5*Y2^2*X1^7-1433*Y1^4*Y2^3*X1^6*X2-2904*Y1^3*Y2^4*X1^5*X2^2-1400*Y1^2*Y2^5*X1^4*X2^3+114*Y1*Y2^6*X1^3*X2^4+10*Y2^7*X1^2*X2^5-937*Y1^3*Y2^3*X1^7-2873*Y1^2*Y2^4*X1^6*X2-2314*Y1*Y2^5*X1^5*X2^2-258*Y2^6*X1^4*X2^3-157*Y1*Y2^4*X1^7-273*Y2^5*X1^6*X2,8*Y1^3*Y2^6*X1*X2^5+5*Y1^2*Y2^7*X2^6-2*Y1^5*Y2^3*X1^5*X2-48*Y1^3*Y2^5*X1^3*X2^3-25*Y1^2*Y2^6*X1^2*X2^4+8*Y1*Y2^7*X1*X2^5+Y2^8*X2^6+16*Y1^4*Y2^3*X1^6+124*Y1^3*Y2^4*X1^5*X2+55*Y1^2*Y2^5*X1^4*X2^2-48*Y1*Y2^6*X1^3*X2^3-5*Y2^7*X1^2*X2^4+101*Y1^2*Y2^4*X1^6+134*Y1*Y2^5*X1^5*X2+11*Y2^6*X1^4*X2^2+17*Y2^5*X1^6,Y1^6*Y2^3*X1^5*X2+32*Y1^3*Y2^6*X1^2*X2^4+20*Y1^2*Y2^7*X1*X2^5+2*Y1*Y2^8*X2^6-8*Y1^5*Y2^3*X1^6-85*Y1^4*Y2^4*X1^5*X2-224*Y1^3*Y2^5*X1^4*X2^2-120*Y1^2*Y2^6*X1^3*X2^3+22*Y1*Y2^7*X1^2*X2^4+4*Y2^8*X1*X2^5-48*Y1^3*Y2^4*X1^6-205*Y1^2*Y2^5*X1^5*X2-202*Y1*Y2^6*X1^4*X2^2-24*Y2^7*X1^3*X2^3-6*Y1*Y2^5*X1^6-23*Y2^6*X1^5*X2 I³∩J: Y1^4*Y2^2*X2^2+Y1^4*Y2*X1^2+8*Y1^3*Y2^2*X1*X2+6*Y1^2*Y2^3*X2^2+6*Y1^2*Y2^2*X1^2+8*Y1*Y2^3*X1*X2+Y2^4*X2^2+Y2^3*X1^2,Y1^5*Y2*X2^2+Y1^5*X1^2+8*Y1^4*Y2*X1*X2+6*Y1^3*Y2^2*X2^2+6*Y1^3*Y2*X1^2+8*Y1^2*Y2^2*X1*X2+Y1*Y2^3*X2^2+Y1*Y2^2*X1^2,Y1^8*X1^4+8*Y1^7*Y2*X1^3*X2-20*Y1^6*Y2*X1^4-200*Y1^5*Y2^2*X1^3*X2-288*Y1^3*Y2^4*X1*X2^3-96*Y1^2*Y2^5*X2^4+230*Y1^4*Y2^2*X1^4+2584*Y1^3*Y2^3*X1^3*X2+1856*Y1^2*Y2^4*X1^2*X2^2-160*Y1*Y2^5*X1*X2^3-16*Y2^6*X2^4+2188*Y1^2*Y2^3*X1^4+2920*Y1*Y2^4*X1^3*X2+352*Y2^5*X1^2*X2^2+369*Y2^4*X1^4,Y1^7*Y2*X1^5*X2+36*Y1^3*Y2^5*X1*X2^5+12*Y1^2*Y2^6*X2^6-8*Y1^6*Y2*X1^6-89*Y1^5*Y2^2*X1^5*X2-472*Y1^3*Y2^4*X1^3*X2^3-220*Y1^2*Y2^5*X1^2*X2^4+20*Y1*Y2^6*X1*X2^5+2*Y2^7*X2^6+240*Y1^4*Y2^2*X1^6+2091*Y1^3*Y2^3*X1^5*X2+1252*Y1^2*Y2^4*X1^4*X2^2-376*Y1*Y2^5*X1^3*X2^3-42*Y2^6*X1^2*X2^4+1732*Y1^2*Y2^3*X1^6+2293*Y1*Y2^4*X1^5*X2+246*Y2^5*X1^4*X2^2+290*Y2^4*X1^6,5*Y1^7*Y2*X1^7+45*Y1^6*Y2^2*X1^6*X2+244*Y1^3*Y2^5*X1^3*X2^4+100*Y1^2*Y2^6*X1^2*X2^5+10*Y1*Y2^7*X1*X2^6+2*Y2^8*X2^7-141*Y1^5*Y2^2*X1^7-1433*Y1^4*Y2^3*X1^6*X2-2904*Y1^3*Y2^4*X1^5*X2^2-1400*Y1^2*Y2^5*X1^4*X2^3+114*Y1*Y2^6*X1^3*X2^4+10*Y2^7*X1^2*X2^5-937*Y1^3*Y2^3*X1^7-2873*Y1^2*Y2^4*X1^6*X2-2314*Y1*Y2^5*X1^5*X2^2-258*Y2^6*X1^4*X2^3-157*Y1*Y2^4*X1^7-273*Y2^5*X1^6*X2,8*Y1^3*Y2^6*X1*X2^5+5*Y1^2*Y2^7*X2^6-2*Y1^5*Y2^3*X1^5*X2-48*Y1^3*Y2^5*X1^3*X2^3-25*Y1^2*Y2^6*X1^2*X2^4+8*Y1*Y2^7*X1*X2^5+Y2^8*X2^6+16*Y1^4*Y2^3*X1^6+124*Y1^3*Y2^4*X1^5*X2+55*Y1^2*Y2^5*X1^4*X2^2-48*Y1*Y2^6*X1^3*X2^3-5*Y2^7*X1^2*X2^4+101*Y1^2*Y2^4*X1^6+134*Y1*Y2^5*X1^5*X2+11*Y2^6*X1^4*X2^2+17*Y2^5*X1^6,Y1^6*Y2^3*X1^5*X2+32*Y1^3*Y2^6*X1^2*X2^4+20*Y1^2*Y2^7*X1*X2^5+2*Y1*Y2^8*X2^6-8*Y1^5*Y2^3*X1^6-85*Y1^4*Y2^4*X1^5*X2-224*Y1^3*Y2^5*X1^4*X2^2-120*Y1^2*Y2^6*X1^3*X2^3+22*Y1*Y2^7*X1^2*X2^4+4*Y2^8*X1*X2^5-48*Y1^3*Y2^4*X1^6-205*Y1^2*Y2^5*X1^5*X2-202*Y1*Y2^6*X1^4*X2^2-24*Y2^7*X1^3*X2^3-6*Y1*Y2^5*X1^6-23*Y2^6*X1^5*X2 I⁴∩J: Y1^4*Y2^3*X2^2+Y1^4*Y2^2*X1^2+8*Y1^3*Y2^3*X1*X2+6*Y1^2*Y2^4*X2^2+6*Y1^2*Y2^3*X1^2+8*Y1*Y2^4*X1*X2+Y2^5*X2^2+Y2^4*X1^2,Y1^5*Y2^2*X2^2+Y1^5*Y2*X1^2+8*Y1^4*Y2^2*X1*X2+6*Y1^3*Y2^3*X2^2+6*Y1^3*Y2^2*X1^2+8*Y1^2*Y2^3*X1*X2+Y1*Y2^4*X2^2+Y1*Y2^3*X1^2,Y1^6*Y2*X2^2+Y1^6*X1^2+8*Y1^5*Y2*X1*X2+6*Y1^4*Y2^2*X2^2+6*Y1^4*Y2*X1^2+8*Y1^3*Y2^2*X1*X2+Y1^2*Y2^3*X2^2+Y1^2*Y2^2*X1^2,Y1^8*X1^4+8*Y1^7*Y2*X1^3*X2-20*Y1^6*Y2*X1^4-200*Y1^5*Y2^2*X1^3*X2-288*Y1^3*Y2^4*X1*X2^3-96*Y1^2*Y2^5*X2^4+230*Y1^4*Y2^2*X1^4+2584*Y1^3*Y2^3*X1^3*X2+1856*Y1^2*Y2^4*X1^2*X2^2-160*Y1*Y2^5*X1*X2^3-16*Y2^6*X2^4+2188*Y1^2*Y2^3*X1^4+2920*Y1*Y2^4*X1^3*X2+352*Y2^5*X1^2*X2^2+369*Y2^4*X1^4,Y1^7*Y2*X1^5*X2+36*Y1^3*Y2^5*X1*X2^5+12*Y1^2*Y2^6*X2^6-8*Y1^6*Y2*X1^6-89*Y1^5*Y2^2*X1^5*X2-472*Y1^3*Y2^4*X1^3*X2^3-220*Y1^2*Y2^5*X1^2*X2^4+20*Y1*Y2^6*X1*X2^5+2*Y2^7*X2^6+240*Y1^4*Y2^2*X1^6+2091*Y1^3*Y2^3*X1^5*X2+1252*Y1^2*Y2^4*X1^4*X2^2-376*Y1*Y2^5*X1^3*X2^3-42*Y2^6*X1^2*X2^4+1732*Y1^2*Y2^3*X1^6+2293*Y1*Y2^4*X1^5*X2+246*Y2^5*X1^4*X2^2+290*Y2^4*X1^6,5*Y1^7*Y2*X1^7+45*Y1^6*Y2^2*X1^6*X2+244*Y1^3*Y2^5*X1^3*X2^4+100*Y1^2*Y2^6*X1^2*X2^5+10*Y1*Y2^7*X1*X2^6+2*Y2^8*X2^7-141*Y1^5*Y2^2*X1^7-1433*Y1^4*Y2^3*X1^6*X2-2904*Y1^3*Y2^4*X1^5*X2^2-1400*Y1^2*Y2^5*X1^4*X2^3+114*Y1*Y2^6*X1^3*X2^4+10*Y2^7*X1^2*X2^5-937*Y1^3*Y2^3*X1^7-2873*Y1^2*Y2^4*X1^6*X2-2314*Y1*Y2^5*X1^5*X2^2-258*Y2^6*X1^4*X2^3-157*Y1*Y2^4*X1^7-273*Y2^5*X1^6*X2,8*Y1^3*Y2^6*X1*X2^5+5*Y1^2*Y2^7*X2^6-2*Y1^5*Y2^3*X1^5*X2-48*Y1^3*Y2^5*X1^3*X2^3-25*Y1^2*Y2^6*X1^2*X2^4+8*Y1*Y2^7*X1*X2^5+Y2^8*X2^6+16*Y1^4*Y2^3*X1^6+124*Y1^3*Y2^4*X1^5*X2+55*Y1^2*Y2^5*X1^4*X2^2-48*Y1*Y2^6*X1^3*X2^3-5*Y2^7*X1^2*X2^4+101*Y1^2*Y2^4*X1^6+134*Y1*Y2^5*X1^5*X2+11*Y2^6*X1^4*X2^2+17*Y2^5*X1^6,Y1^6*Y2^3*X1^5*X2+32*Y1^3*Y2^6*X1^2*X2^4+20*Y1^2*Y2^7*X1*X2^5+2*Y1*Y2^8*X2^6-8*Y1^5*Y2^3*X1^6-85*Y1^4*Y2^4*X1^5*X2-224*Y1^3*Y2^5*X1^4*X2^2-120*Y1^2*Y2^6*X1^3*X2^3+22*Y1*Y2^7*X1^2*X2^4+4*Y2^8*X1*X2^5-48*Y1^3*Y2^4*X1^6-205*Y1^2*Y2^5*X1^5*X2-202*Y1*Y2^6*X1^4*X2^2-24*Y2^7*X1^3*X2^3-6*Y1*Y2^5*X1^6-23*Y2^6*X1^5*X2 I⁵∩J: 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I⁹∩J: 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In [51]:
arr2 = [Y1^4*Y2*X2^2+Y1^4*X1^2+8*Y1^3*Y2*X1*X2+6*Y1^2*Y2^2*X2^2+6*Y1^2*Y2*X1^2+8*Y1*Y2^2*X1*X2+Y2^3*X2^2+Y2^2*X1^2,Y1^8*X1^4+8*Y1^7*Y2*X1^3*X2-20*Y1^6*Y2*X1^4-200*Y1^5*Y2^2*X1^3*X2-288*Y1^3*Y2^4*X1*X2^3-96*Y1^2*Y2^5*X2^4+230*Y1^4*Y2^2*X1^4+2584*Y1^3*Y2^3*X1^3*X2+1856*Y1^2*Y2^4*X1^2*X2^2-160*Y1*Y2^5*X1*X2^3-16*Y2^6*X2^4+2188*Y1^2*Y2^3*X1^4+2920*Y1*Y2^4*X1^3*X2+352*Y2^5*X1^2*X2^2+369*Y2^4*X1^4,Y1^7*Y2*X1^5*X2+36*Y1^3*Y2^5*X1*X2^5+12*Y1^2*Y2^6*X2^6-8*Y1^6*Y2*X1^6-89*Y1^5*Y2^2*X1^5*X2-472*Y1^3*Y2^4*X1^3*X2^3-220*Y1^2*Y2^5*X1^2*X2^4+20*Y1*Y2^6*X1*X2^5+2*Y2^7*X2^6+240*Y1^4*Y2^2*X1^6+2091*Y1^3*Y2^3*X1^5*X2+1252*Y1^2*Y2^4*X1^4*X2^2-376*Y1*Y2^5*X1^3*X2^3-42*Y2^6*X1^2*X2^4+1732*Y1^2*Y2^3*X1^6+2293*Y1*Y2^4*X1^5*X2+246*Y2^5*X1^4*X2^2+290*Y2^4*X1^6,5*Y1^7*Y2*X1^7+45*Y1^6*Y2^2*X1^6*X2+244*Y1^3*Y2^5*X1^3*X2^4+100*Y1^2*Y2^6*X1^2*X2^5+10*Y1*Y2^7*X1*X2^6+2*Y2^8*X2^7-141*Y1^5*Y2^2*X1^7-1433*Y1^4*Y2^3*X1^6*X2-2904*Y1^3*Y2^4*X1^5*X2^2-1400*Y1^2*Y2^5*X1^4*X2^3+114*Y1*Y2^6*X1^3*X2^4+10*Y2^7*X1^2*X2^5-937*Y1^3*Y2^3*X1^7-2873*Y1^2*Y2^4*X1^6*X2-2314*Y1*Y2^5*X1^5*X2^2-258*Y2^6*X1^4*X2^3-157*Y1*Y2^4*X1^7-273*Y2^5*X1^6*X2,8*Y1^3*Y2^6*X1*X2^5+5*Y1^2*Y2^7*X2^6-2*Y1^5*Y2^3*X1^5*X2-48*Y1^3*Y2^5*X1^3*X2^3-25*Y1^2*Y2^6*X1^2*X2^4+8*Y1*Y2^7*X1*X2^5+Y2^8*X2^6+16*Y1^4*Y2^3*X1^6+124*Y1^3*Y2^4*X1^5*X2+55*Y1^2*Y2^5*X1^4*X2^2-48*Y1*Y2^6*X1^3*X2^3-5*Y2^7*X1^2*X2^4+101*Y1^2*Y2^4*X1^6+134*Y1*Y2^5*X1^5*X2+11*Y2^6*X1^4*X2^2+17*Y2^5*X1^6,Y1^6*Y2^3*X1^5*X2+32*Y1^3*Y2^6*X1^2*X2^4+20*Y1^2*Y2^7*X1*X2^5+2*Y1*Y2^8*X2^6-8*Y1^5*Y2^3*X1^6-85*Y1^4*Y2^4*X1^5*X2-224*Y1^3*Y2^5*X1^4*X2^2-120*Y1^2*Y2^6*X1^3*X2^3+22*Y1*Y2^7*X1^2*X2^4+4*Y2^8*X1*X2^5-48*Y1^3*Y2^4*X1^6-205*Y1^2*Y2^5*X1^5*X2-202*Y1*Y2^6*X1^4*X2^2-24*Y2^7*X1^3*X2^3-6*Y1*Y2^5*X1^6-23*Y2^6*X1^5*X2]
arr3 = [Y1^4*Y2^2*X2^2+Y1^4*Y2*X1^2+8*Y1^3*Y2^2*X1*X2+6*Y1^2*Y2^3*X2^2+6*Y1^2*Y2^2*X1^2+8*Y1*Y2^3*X1*X2+Y2^4*X2^2+Y2^3*X1^2,Y1^5*Y2*X2^2+Y1^5*X1^2+8*Y1^4*Y2*X1*X2+6*Y1^3*Y2^2*X2^2+6*Y1^3*Y2*X1^2+8*Y1^2*Y2^2*X1*X2+Y1*Y2^3*X2^2+Y1*Y2^2*X1^2,Y1^8*X1^4+8*Y1^7*Y2*X1^3*X2-20*Y1^6*Y2*X1^4-200*Y1^5*Y2^2*X1^3*X2-288*Y1^3*Y2^4*X1*X2^3-96*Y1^2*Y2^5*X2^4+230*Y1^4*Y2^2*X1^4+2584*Y1^3*Y2^3*X1^3*X2+1856*Y1^2*Y2^4*X1^2*X2^2-160*Y1*Y2^5*X1*X2^3-16*Y2^6*X2^4+2188*Y1^2*Y2^3*X1^4+2920*Y1*Y2^4*X1^3*X2+352*Y2^5*X1^2*X2^2+369*Y2^4*X1^4,Y1^7*Y2*X1^5*X2+36*Y1^3*Y2^5*X1*X2^5+12*Y1^2*Y2^6*X2^6-8*Y1^6*Y2*X1^6-89*Y1^5*Y2^2*X1^5*X2-472*Y1^3*Y2^4*X1^3*X2^3-220*Y1^2*Y2^5*X1^2*X2^4+20*Y1*Y2^6*X1*X2^5+2*Y2^7*X2^6+240*Y1^4*Y2^2*X1^6+2091*Y1^3*Y2^3*X1^5*X2+1252*Y1^2*Y2^4*X1^4*X2^2-376*Y1*Y2^5*X1^3*X2^3-42*Y2^6*X1^2*X2^4+1732*Y1^2*Y2^3*X1^6+2293*Y1*Y2^4*X1^5*X2+246*Y2^5*X1^4*X2^2+290*Y2^4*X1^6,5*Y1^7*Y2*X1^7+45*Y1^6*Y2^2*X1^6*X2+244*Y1^3*Y2^5*X1^3*X2^4+100*Y1^2*Y2^6*X1^2*X2^5+10*Y1*Y2^7*X1*X2^6+2*Y2^8*X2^7-141*Y1^5*Y2^2*X1^7-1433*Y1^4*Y2^3*X1^6*X2-2904*Y1^3*Y2^4*X1^5*X2^2-1400*Y1^2*Y2^5*X1^4*X2^3+114*Y1*Y2^6*X1^3*X2^4+10*Y2^7*X1^2*X2^5-937*Y1^3*Y2^3*X1^7-2873*Y1^2*Y2^4*X1^6*X2-2314*Y1*Y2^5*X1^5*X2^2-258*Y2^6*X1^4*X2^3-157*Y1*Y2^4*X1^7-273*Y2^5*X1^6*X2,8*Y1^3*Y2^6*X1*X2^5+5*Y1^2*Y2^7*X2^6-2*Y1^5*Y2^3*X1^5*X2-48*Y1^3*Y2^5*X1^3*X2^3-25*Y1^2*Y2^6*X1^2*X2^4+8*Y1*Y2^7*X1*X2^5+Y2^8*X2^6+16*Y1^4*Y2^3*X1^6+124*Y1^3*Y2^4*X1^5*X2+55*Y1^2*Y2^5*X1^4*X2^2-48*Y1*Y2^6*X1^3*X2^3-5*Y2^7*X1^2*X2^4+101*Y1^2*Y2^4*X1^6+134*Y1*Y2^5*X1^5*X2+11*Y2^6*X1^4*X2^2+17*Y2^5*X1^6,Y1^6*Y2^3*X1^5*X2+32*Y1^3*Y2^6*X1^2*X2^4+20*Y1^2*Y2^7*X1*X2^5+2*Y1*Y2^8*X2^6-8*Y1^5*Y2^3*X1^6-85*Y1^4*Y2^4*X1^5*X2-224*Y1^3*Y2^5*X1^4*X2^2-120*Y1^2*Y2^6*X1^3*X2^3+22*Y1*Y2^7*X1^2*X2^4+4*Y2^8*X1*X2^5-48*Y1^3*Y2^4*X1^6-205*Y1^2*Y2^5*X1^5*X2-202*Y1*Y2^6*X1^4*X2^2-24*Y2^7*X1^3*X2^3-6*Y1*Y2^5*X1^6-23*Y2^6*X1^5*X2]
arr4 = [Y1^4*Y2^3*X2^2+Y1^4*Y2^2*X1^2+8*Y1^3*Y2^3*X1*X2+6*Y1^2*Y2^4*X2^2+6*Y1^2*Y2^3*X1^2+8*Y1*Y2^4*X1*X2+Y2^5*X2^2+Y2^4*X1^2,Y1^5*Y2^2*X2^2+Y1^5*Y2*X1^2+8*Y1^4*Y2^2*X1*X2+6*Y1^3*Y2^3*X2^2+6*Y1^3*Y2^2*X1^2+8*Y1^2*Y2^3*X1*X2+Y1*Y2^4*X2^2+Y1*Y2^3*X1^2,Y1^6*Y2*X2^2+Y1^6*X1^2+8*Y1^5*Y2*X1*X2+6*Y1^4*Y2^2*X2^2+6*Y1^4*Y2*X1^2+8*Y1^3*Y2^2*X1*X2+Y1^2*Y2^3*X2^2+Y1^2*Y2^2*X1^2,Y1^8*X1^4+8*Y1^7*Y2*X1^3*X2-20*Y1^6*Y2*X1^4-200*Y1^5*Y2^2*X1^3*X2-288*Y1^3*Y2^4*X1*X2^3-96*Y1^2*Y2^5*X2^4+230*Y1^4*Y2^2*X1^4+2584*Y1^3*Y2^3*X1^3*X2+1856*Y1^2*Y2^4*X1^2*X2^2-160*Y1*Y2^5*X1*X2^3-16*Y2^6*X2^4+2188*Y1^2*Y2^3*X1^4+2920*Y1*Y2^4*X1^3*X2+352*Y2^5*X1^2*X2^2+369*Y2^4*X1^4,Y1^7*Y2*X1^5*X2+36*Y1^3*Y2^5*X1*X2^5+12*Y1^2*Y2^6*X2^6-8*Y1^6*Y2*X1^6-89*Y1^5*Y2^2*X1^5*X2-472*Y1^3*Y2^4*X1^3*X2^3-220*Y1^2*Y2^5*X1^2*X2^4+20*Y1*Y2^6*X1*X2^5+2*Y2^7*X2^6+240*Y1^4*Y2^2*X1^6+2091*Y1^3*Y2^3*X1^5*X2+1252*Y1^2*Y2^4*X1^4*X2^2-376*Y1*Y2^5*X1^3*X2^3-42*Y2^6*X1^2*X2^4+1732*Y1^2*Y2^3*X1^6+2293*Y1*Y2^4*X1^5*X2+246*Y2^5*X1^4*X2^2+290*Y2^4*X1^6,5*Y1^7*Y2*X1^7+45*Y1^6*Y2^2*X1^6*X2+244*Y1^3*Y2^5*X1^3*X2^4+100*Y1^2*Y2^6*X1^2*X2^5+10*Y1*Y2^7*X1*X2^6+2*Y2^8*X2^7-141*Y1^5*Y2^2*X1^7-1433*Y1^4*Y2^3*X1^6*X2-2904*Y1^3*Y2^4*X1^5*X2^2-1400*Y1^2*Y2^5*X1^4*X2^3+114*Y1*Y2^6*X1^3*X2^4+10*Y2^7*X1^2*X2^5-937*Y1^3*Y2^3*X1^7-2873*Y1^2*Y2^4*X1^6*X2-2314*Y1*Y2^5*X1^5*X2^2-258*Y2^6*X1^4*X2^3-157*Y1*Y2^4*X1^7-273*Y2^5*X1^6*X2,8*Y1^3*Y2^6*X1*X2^5+5*Y1^2*Y2^7*X2^6-2*Y1^5*Y2^3*X1^5*X2-48*Y1^3*Y2^5*X1^3*X2^3-25*Y1^2*Y2^6*X1^2*X2^4+8*Y1*Y2^7*X1*X2^5+Y2^8*X2^6+16*Y1^4*Y2^3*X1^6+124*Y1^3*Y2^4*X1^5*X2+55*Y1^2*Y2^5*X1^4*X2^2-48*Y1*Y2^6*X1^3*X2^3-5*Y2^7*X1^2*X2^4+101*Y1^2*Y2^4*X1^6+134*Y1*Y2^5*X1^5*X2+11*Y2^6*X1^4*X2^2+17*Y2^5*X1^6,Y1^6*Y2^3*X1^5*X2+32*Y1^3*Y2^6*X1^2*X2^4+20*Y1^2*Y2^7*X1*X2^5+2*Y1*Y2^8*X2^6-8*Y1^5*Y2^3*X1^6-85*Y1^4*Y2^4*X1^5*X2-224*Y1^3*Y2^5*X1^4*X2^2-120*Y1^2*Y2^6*X1^3*X2^3+22*Y1*Y2^7*X1^2*X2^4+4*Y2^8*X1*X2^5-48*Y1^3*Y2^4*X1^6-205*Y1^2*Y2^5*X1^5*X2-202*Y1*Y2^6*X1^4*X2^2-24*Y2^7*X1^3*X2^3-6*Y1*Y2^5*X1^6-23*Y2^6*X1^5*X2]
arr5 = [Y1^4*Y2^4*X2^2+Y1^4*Y2^3*X1^2+8*Y1^3*Y2^4*X1*X2+6*Y1^2*Y2^5*X2^2+6*Y1^2*Y2^4*X1^2+8*Y1*Y2^5*X1*X2+Y2^6*X2^2+Y2^5*X1^2,Y1^5*Y2^3*X2^2+Y1^5*Y2^2*X1^2+8*Y1^4*Y2^3*X1*X2+6*Y1^3*Y2^4*X2^2+6*Y1^3*Y2^3*X1^2+8*Y1^2*Y2^4*X1*X2+Y1*Y2^5*X2^2+Y1*Y2^4*X1^2,Y1^6*Y2^2*X2^2+Y1^6*Y2*X1^2+8*Y1^5*Y2^2*X1*X2+6*Y1^4*Y2^3*X2^2+6*Y1^4*Y2^2*X1^2+8*Y1^3*Y2^3*X1*X2+Y1^2*Y2^4*X2^2+Y1^2*Y2^3*X1^2,Y1^7*Y2*X2^2+Y1^7*X1^2+8*Y1^6*Y2*X1*X2+6*Y1^5*Y2^2*X2^2+6*Y1^5*Y2*X1^2+8*Y1^4*Y2^2*X1*X2+Y1^3*Y2^3*X2^2+Y1^3*Y2^2*X1^2,Y1^8*X1^4+8*Y1^7*Y2*X1^3*X2-20*Y1^6*Y2*X1^4-200*Y1^5*Y2^2*X1^3*X2-369*Y1^4*Y2^3*X1^2*X2^2-288*Y1^3*Y2^4*X1*X2^3-96*Y1^2*Y2^5*X2^4-139*Y1^4*Y2^2*X1^4-368*Y1^3*Y2^3*X1^3*X2-358*Y1^2*Y2^4*X1^2*X2^2-160*Y1*Y2^5*X1*X2^3-16*Y2^6*X2^4-26*Y1^2*Y2^3*X1^4-32*Y1*Y2^4*X1^3*X2-17*Y2^5*X1^2*X2^2,Y1^7*Y2*X1^5*X2+36*Y1^3*Y2^5*X1*X2^5+12*Y1^2*Y2^6*X2^6-8*Y1^6*Y2*X1^6-89*Y1^5*Y2^2*X1^5*X2-290*Y1^4*Y2^3*X1^4*X2^2-472*Y1^3*Y2^4*X1^3*X2^3-220*Y1^2*Y2^5*X1^2*X2^4+20*Y1*Y2^6*X1*X2^5+2*Y2^7*X2^6-50*Y1^4*Y2^2*X1^6-229*Y1^3*Y2^3*X1^5*X2-488*Y1^2*Y2^4*X1^4*X2^2-376*Y1*Y2^5*X1^3*X2^3-42*Y2^6*X1^2*X2^4-8*Y1^2*Y2^3*X1^6-27*Y1*Y2^4*X1^5*X2-44*Y2^5*X1^4*X2^2,5*Y1^7*Y2*X1^7+45*Y1^6*Y2^2*X1^6*X2+244*Y1^3*Y2^5*X1^3*X2^4+100*Y1^2*Y2^6*X1^2*X2^5+10*Y1*Y2^7*X1*X2^6+2*Y2^8*X2^7-141*Y1^5*Y2^2*X1^7-1433*Y1^4*Y2^3*X1^6*X2-2904*Y1^3*Y2^4*X1^5*X2^2-1400*Y1^2*Y2^5*X1^4*X2^3+114*Y1*Y2^6*X1^3*X2^4+10*Y2^7*X1^2*X2^5-937*Y1^3*Y2^3*X1^7-2873*Y1^2*Y2^4*X1^6*X2-2314*Y1*Y2^5*X1^5*X2^2-258*Y2^6*X1^4*X2^3-157*Y1*Y2^4*X1^7-273*Y2^5*X1^6*X2,8*Y1^3*Y2^6*X1*X2^5+5*Y1^2*Y2^7*X2^6-2*Y1^5*Y2^3*X1^5*X2-48*Y1^3*Y2^5*X1^3*X2^3-25*Y1^2*Y2^6*X1^2*X2^4+8*Y1*Y2^7*X1*X2^5+Y2^8*X2^6+16*Y1^4*Y2^3*X1^6+124*Y1^3*Y2^4*X1^5*X2+55*Y1^2*Y2^5*X1^4*X2^2-48*Y1*Y2^6*X1^3*X2^3-5*Y2^7*X1^2*X2^4+101*Y1^2*Y2^4*X1^6+134*Y1*Y2^5*X1^5*X2+11*Y2^6*X1^4*X2^2+17*Y2^5*X1^6,Y1^6*Y2^3*X1^5*X2+32*Y1^3*Y2^6*X1^2*X2^4+20*Y1^2*Y2^7*X1*X2^5+2*Y1*Y2^8*X2^6-8*Y1^5*Y2^3*X1^6-85*Y1^4*Y2^4*X1^5*X2-224*Y1^3*Y2^5*X1^4*X2^2-120*Y1^2*Y2^6*X1^3*X2^3+22*Y1*Y2^7*X1^2*X2^4+4*Y2^8*X1*X2^5-48*Y1^3*Y2^4*X1^6-205*Y1^2*Y2^5*X1^5*X2-202*Y1*Y2^6*X1^4*X2^2-24*Y2^7*X1^3*X2^3-6*Y1*Y2^5*X1^6-23*Y2^6*X1^5*X2]
arr6 = [Y1^4*Y2^5*X2^2+Y1^4*Y2^4*X1^2+8*Y1^3*Y2^5*X1*X2+6*Y1^2*Y2^6*X2^2+6*Y1^2*Y2^5*X1^2+8*Y1*Y2^6*X1*X2+Y2^7*X2^2+Y2^6*X1^2,Y1^5*Y2^4*X2^2+Y1^5*Y2^3*X1^2+8*Y1^4*Y2^4*X1*X2+6*Y1^3*Y2^5*X2^2+6*Y1^3*Y2^4*X1^2+8*Y1^2*Y2^5*X1*X2+Y1*Y2^6*X2^2+Y1*Y2^5*X1^2,Y1^6*Y2^3*X2^2+Y1^6*Y2^2*X1^2+8*Y1^5*Y2^3*X1*X2+6*Y1^4*Y2^4*X2^2+6*Y1^4*Y2^3*X1^2+8*Y1^3*Y2^4*X1*X2+Y1^2*Y2^5*X2^2+Y1^2*Y2^4*X1^2,Y1^7*Y2^2*X2^2+Y1^7*Y2*X1^2+8*Y1^6*Y2^2*X1*X2+6*Y1^5*Y2^3*X2^2+6*Y1^5*Y2^2*X1^2+8*Y1^4*Y2^3*X1*X2+Y1^3*Y2^4*X2^2+Y1^3*Y2^3*X1^2,Y1^8*Y2*X2^2+Y1^8*X1^2+8*Y1^7*Y2*X1*X2+6*Y1^6*Y2^2*X2^2+6*Y1^6*Y2*X1^2+8*Y1^5*Y2^2*X1*X2+Y1^4*Y2^3*X2^2+Y1^4*Y2^2*X1^2,Y1^8*X1^4+8*Y1^7*Y2*X1^3*X2+26*Y1^6*Y2^2*X1^2*X2^2+32*Y1^5*Y2^3*X1*X2^3+17*Y1^4*Y2^4*X2^4+6*Y1^6*Y2*X1^4+40*Y1^5*Y2^2*X1^3*X2+60*Y1^4*Y2^3*X1^2*X2^2+40*Y1^3*Y2^4*X1*X2^3+6*Y1^2*Y2^5*X2^4+17*Y1^4*Y2^2*X1^4+32*Y1^3*Y2^3*X1^3*X2+26*Y1^2*Y2^4*X1^2*X2^2+8*Y1*Y2^5*X1*X2^3+Y2^6*X2^4,Y1^7*Y2*X1^5*X2+8*Y1^6*Y2^2*X1^4*X2^2+27*Y1^5*Y2^3*X1^3*X2^3+44*Y1^4*Y2^4*X1^2*X2^4+36*Y1^3*Y2^5*X1*X2^5+12*Y1^2*Y2^6*X2^6+2*Y1^5*Y2^2*X1^5*X2+18*Y1^4*Y2^3*X1^4*X2^2+42*Y1^3*Y2^4*X1^3*X2^3+44*Y1^2*Y2^5*X1^2*X2^4+20*Y1*Y2^6*X1*X2^5+2*Y2^7*X2^6-2*Y1^4*Y2^2*X1^6-3*Y1^3*Y2^3*X1^5*X2+3*Y1*Y2^5*X1^3*X2^3+2*Y2^6*X1^2*X2^4,5*Y1^7*Y2*X1^7+45*Y1^6*Y2^2*X1^6*X2+157*Y1^5*Y2^3*X1^5*X2^2+273*Y1^4*Y2^4*X1^4*X2^3+244*Y1^3*Y2^5*X1^3*X2^4+100*Y1^2*Y2^6*X1^2*X2^5+10*Y1*Y2^7*X1*X2^6+2*Y2^8*X2^7+16*Y1^5*Y2^2*X1^7+96*Y1^4*Y2^3*X1^6*X2+222*Y1^3*Y2^4*X1^5*X2^2+238*Y1^2*Y2^5*X1^4*X2^3+114*Y1*Y2^6*X1^3*X2^4+10*Y2^7*X1^2*X2^5+5*Y1^3*Y2^3*X1^7+21*Y1^2*Y2^4*X1^6*X2+27*Y1*Y2^5*X1^5*X2^2+15*Y2^6*X1^4*X2^3,8*Y1^3*Y2^6*X1*X2^5+5*Y1^2*Y2^7*X2^6-2*Y1^5*Y2^3*X1^5*X2-17*Y1^4*Y2^4*X1^4*X2^2-48*Y1^3*Y2^5*X1^3*X2^3-25*Y1^2*Y2^6*X1^2*X2^4+8*Y1*Y2^7*X1*X2^5+Y2^8*X2^6-Y1^4*Y2^3*X1^6-12*Y1^3*Y2^4*X1^5*X2-47*Y1^2*Y2^5*X1^4*X2^2-48*Y1*Y2^6*X1^3*X2^3-5*Y2^7*X1^2*X2^4-Y1^2*Y2^4*X1^6-2*Y1*Y2^5*X1^5*X2-6*Y2^6*X1^4*X2^2,Y1^6*Y2^3*X1^5*X2+32*Y1^3*Y2^6*X1^2*X2^4+20*Y1^2*Y2^7*X1*X2^5+2*Y1*Y2^8*X2^6-8*Y1^5*Y2^3*X1^6-85*Y1^4*Y2^4*X1^5*X2-224*Y1^3*Y2^5*X1^4*X2^2-120*Y1^2*Y2^6*X1^3*X2^3+22*Y1*Y2^7*X1^2*X2^4+4*Y2^8*X1*X2^5-48*Y1^3*Y2^4*X1^6-205*Y1^2*Y2^5*X1^5*X2-202*Y1*Y2^6*X1^4*X2^2-24*Y2^7*X1^3*X2^3-6*Y1*Y2^5*X1^6-23*Y2^6*X1^5*X2]
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arr8 = 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2^9*X2^9+12*Y1^7*Y2*X1^9+84*Y1^6*Y2^2*X1^8*X2+252*Y1^5*Y2^3*X1^7*X2^2+420*Y1^4*Y2^4*X1^6*X2^3+420*Y1^3*Y2^5*X1^5*X2^4+252*Y1^2*Y2^6*X1^4*X2^5+84*Y1*Y2^7*X1^3*X2^6+12*Y2^8*X1^2*X2^7]
arr9 = [Y1^4*Y2^8*X2^2+Y1^4*Y2^7*X1^2+8*Y1^3*Y2^8*X1*X2+6*Y1^2*Y2^9*X2^2+6*Y1^2*Y2^8*X1^2+8*Y1*Y2^9*X1*X2+Y2^10*X2^2+Y2^9*X1^2,Y1^5*Y2^7*X2^2+Y1^5*Y2^6*X1^2+8*Y1^4*Y2^7*X1*X2+6*Y1^3*Y2^8*X2^2+6*Y1^3*Y2^7*X1^2+8*Y1^2*Y2^8*X1*X2+Y1*Y2^9*X2^2+Y1*Y2^8*X1^2,Y1^6*Y2^6*X2^2+Y1^6*Y2^5*X1^2+8*Y1^5*Y2^6*X1*X2+6*Y1^4*Y2^7*X2^2+6*Y1^4*Y2^6*X1^2+8*Y1^3*Y2^7*X1*X2+Y1^2*Y2^8*X2^2+Y1^2*Y2^7*X1^2,Y1^7*Y2^5*X2^2+Y1^7*Y2^4*X1^2+8*Y1^6*Y2^5*X1*X2+6*Y1^5*Y2^6*X2^2+6*Y1^5*Y2^5*X1^2+8*Y1^4*Y2^6*X1*X2+Y1^3*Y2^7*X2^2+Y1^3*Y2^6*X1^2,Y1^8*Y2^4*X2^2+Y1^8*Y2^3*X1^2+8*Y1^7*Y2^4*X1*X2+6*Y1^6*Y2^5*X2^2+6*Y1^6*Y2^4*X1^2+8*Y1^5*Y2^5*X1*X2+Y1^4*Y2^6*X2^2+Y1^4*Y2^5*X1^2,Y1^9*Y2^3*X2^2+Y1^9*Y2^2*X1^2+8*Y1^8*Y2^3*X1*X2+6*Y1^7*Y2^4*X2^2+6*Y1^7*Y2^3*X1^2+8*Y1^6*Y2^4*X1*X2+Y1^5*Y2^5*X2^2+Y1^5*Y2^4*X1^2,Y1^10*Y2^2*X2^2+Y1^10*Y2*X1^2+8*Y1^9*Y2^2*X1*X2+6*Y1^8*Y2^3*X2^2+6*Y1^8*Y2^2*X1^2+8*Y1^7*Y2^3*X1*X2+Y1^6*Y2^4*X2^2+Y1^6*Y2^3*X1^2,Y1^11*Y2*X2^2+Y1^11*X1^2+8*Y1^10*Y2*X1*X2+6*Y1^9*Y2^2*X2^2+6*Y1^9*Y2*X1^2+8*Y1^8*Y2^2*X1*X2+Y1^7*Y2^3*X2^2+Y1^7*Y2^2*X1^2,Y1^8*Y2^3*X1^4+8*Y1^7*Y2^4*X1^3*X2+26*Y1^6*Y2^5*X1^2*X2^2+32*Y1^5*Y2^6*X1*X2^3+17*Y1^4*Y2^7*X2^4+6*Y1^6*Y2^4*X1^4+40*Y1^5*Y2^5*X1^3*X2+60*Y1^4*Y2^6*X1^2*X2^2+40*Y1^3*Y2^7*X1*X2^3+6*Y1^2*Y2^8*X2^4+17*Y1^4*Y2^5*X1^4+32*Y1^3*Y2^6*X1^3*X2+26*Y1^2*Y2^7*X1^2*X2^2+8*Y1*Y2^8*X1*X2^3+Y2^9*X2^4,Y1^9*Y2^2*X1^4+8*Y1^8*Y2^3*X1^3*X2+26*Y1^7*Y2^4*X1^2*X2^2+32*Y1^6*Y2^5*X1*X2^3+17*Y1^5*Y2^6*X2^4+6*Y1^7*Y2^3*X1^4+40*Y1^6*Y2^4*X1^3*X2+60*Y1^5*Y2^5*X1^2*X2^2+40*Y1^4*Y2^6*X1*X2^3+6*Y1^3*Y2^7*X2^4+17*Y1^5*Y2^4*X1^4+32*Y1^4*Y2^5*X1^3*X2+26*Y1^3*Y2^6*X1^2*X2^2+8*Y1^2*Y2^7*X1*X2^3+Y1*Y2^8*X2^4,Y1^10*Y2*X1^4+8*Y1^9*Y2^2*X1^3*X2+26*Y1^8*Y2^3*X1^2*X2^2+32*Y1^7*Y2^4*X1*X2^3+17*Y1^6*Y2^5*X2^4+6*Y1^8*Y2^2*X1^4+40*Y1^7*Y2^3*X1^3*X2+60*Y1^6*Y2^4*X1^2*X2^2+40*Y1^5*Y2^5*X1*X2^3+6*Y1^4*Y2^6*X2^4+17*Y1^6*Y2^3*X1^4+32*Y1^5*Y2^4*X1^3*X2+26*Y1^4*Y2^5*X1^2*X2^2+8*Y1^3*Y2^6*X1*X2^3+Y1^2*Y2^7*X2^4,Y1^11*X1^4+8*Y1^10*Y2*X1^3*X2+26*Y1^9*Y2^2*X1^2*X2^2+32*Y1^8*Y2^3*X1*X2^3+17*Y1^7*Y2^4*X2^4+6*Y1^9*Y2*X1^4+40*Y1^8*Y2^2*X1^3*X2+60*Y1^7*Y2^3*X1^2*X2^2+40*Y1^6*Y2^4*X1*X2^3+6*Y1^5*Y2^5*X2^4+17*Y1^7*Y2^2*X1^4+32*Y1^6*Y2^3*X1^3*X2+26*Y1^5*Y2^4*X1^2*X2^2+8*Y1^4*Y2^5*X1*X2^3+Y1^3*Y2^6*X2^4,52*Y1^6*Y2^5*X1^2*X2^4+90*Y1^5*Y2^6*X1*X2^5+60*Y1^4*Y2^7*X2^6+Y1^8*Y2^2*X1^6-16*Y1^7*Y2^3*X1^5*X2-127*Y1^6*Y2^4*X1^4*X2^2-136*Y1^5*Y2^5*X1^3*X2^3-25*Y1^4*Y2^6*X1^2*X2^4+52*Y1^3*Y2^7*X1*X2^5+7*Y1^2*Y2^8*X2^6-7*Y1^6*Y2^3*X1^6-112*Y1^5*Y2^4*X1^5*X2-203*Y1^4*Y2^5*X1^4*X2^2-168*Y1^3*Y2^6*X1^3*X2^3-61*Y1^2*Y2^7*X1^2*X2^4-14*Y1*Y2^8*X1*X2^5-Y2^9*X2^6,Y1^6*Y2^5*X1^3*X2^3+12*Y1^5*Y2^6*X1^2*X2^4+19*Y1^4*Y2^7*X1*X2^5+12*Y1^3*Y2^8*X2^6-4*Y1^6*Y2^4*X1^5*X2-22*Y1^5*Y2^5*X1^4*X2^2-18*Y1^4*Y2^6*X1^3*X2^3+4*Y1^3*Y2^7*X1^2*X2^4+14*Y1^2*Y2^8*X1*X2^5+2*Y1*Y2^9*X2^6-2*Y1^5*Y2^4*X1^6-20*Y1^4*Y2^5*X1^5*X2-32*Y1^3*Y2^6*X1^4*X2^2-23*Y1^2*Y2^7*X1^3*X2^3-6*Y1*Y2^8*X1^2*X2^4-Y2^9*X1*X2^5,Y1^6*Y2^5*X1^4*X2^2+2*Y1^5*Y2^6*X1^3*X2^3+6*Y1^4*Y2^7*X1^2*X2^4+8*Y1^3*Y2^8*X1*X2^5+5*Y1^2*Y2^9*X2^6+Y1^6*Y2^4*X1^6+8*Y1^5*Y2^5*X1^5*X2+11*Y1^4*Y2^6*X1^4*X2^2+12*Y1^3*Y2^7*X1^3*X2^3+11*Y1^2*Y2^8*X1^2*X2^4+8*Y1*Y2^9*X1*X2^5+Y2^10*X2^6+5*Y1^4*Y2^5*X1^6+8*Y1^3*Y2^6*X1^5*X2+6*Y1^2*Y2^7*X1^4*X2^2+2*Y1*Y2^8*X1^3*X2^3+Y2^9*X1^2*X2^4,Y1^6*Y2^5*X1^5*X2+6*Y1^5*Y2^6*X1^4*X2^2+23*Y1^4*Y2^7*X1^3*X2^3+32*Y1^3*Y2^8*X1^2*X2^4+20*Y1^2*Y2^9*X1*X2^5+2*Y1*Y2^10*X2^6-2*Y1^5*Y2^5*X1^6-14*Y1^4*Y2^6*X1^5*X2-4*Y1^3*Y2^7*X1^4*X2^2+18*Y1^2*Y2^8*X1^3*X2^3+22*Y1*Y2^9*X1^2*X2^4+4*Y2^10*X1*X2^5-12*Y1^3*Y2^6*X1^6-19*Y1^2*Y2^7*X1^5*X2-12*Y1*Y2^8*X1^4*X2^2-Y2^9*X1^3*X2^3,52*Y1^7*Y2^4*X1^2*X2^4+90*Y1^6*Y2^5*X1*X2^5+60*Y1^5*Y2^6*X2^6+Y1^9*Y2*X1^6-16*Y1^8*Y2^2*X1^5*X2-127*Y1^7*Y2^3*X1^4*X2^2-136*Y1^6*Y2^4*X1^3*X2^3-25*Y1^5*Y2^5*X1^2*X2^4+52*Y1^4*Y2^6*X1*X2^5+7*Y1^3*Y2^7*X2^6-7*Y1^7*Y2^2*X1^6-112*Y1^6*Y2^3*X1^5*X2-203*Y1^5*Y2^4*X1^4*X2^2-168*Y1^4*Y2^5*X1^3*X2^3-61*Y1^3*Y2^6*X1^2*X2^4-14*Y1^2*Y2^7*X1*X2^5-Y1*Y2^8*X2^6,13*Y1^7*Y2^4*X1^3*X2^3-23*Y1^5*Y2^6*X1*X2^5-24*Y1^4*Y2^7*X2^6-3*Y1^8*Y2^2*X1^6-4*Y1^7*Y2^3*X1^5*X2+95*Y1^6*Y2^4*X1^4*X2^2+174*Y1^5*Y2^5*X1^3*X2^3+127*Y1^4*Y2^6*X1^2*X2^4+26*Y1^3*Y2^7*X1*X2^5+5*Y1^2*Y2^8*X2^6-5*Y1^6*Y2^3*X1^6+76*Y1^5*Y2^4*X1^5*X2+193*Y1^4*Y2^5*X1^4*X2^2+205*Y1^3*Y2^6*X1^3*X2^3+105*Y1^2*Y2^7*X1^2*X2^4+29*Y1*Y2^8*X1*X2^5+3*Y2^9*X2^6,Y1^7*Y2^4*X1^4*X2^2-18*Y1^5*Y2^6*X1^2*X2^4-30*Y1^4*Y2^7*X1*X2^5-19*Y1^3*Y2^8*X2^6+Y1^7*Y2^3*X1^6+16*Y1^6*Y2^4*X1^5*X2+55*Y1^5*Y2^5*X1^4*X2^2+48*Y1^4*Y2^6*X1^3*X2^3+3*Y1^3*Y2^7*X1^2*X2^4-20*Y1^2*Y2^8*X1*X2^5-3*Y1*Y2^9*X2^6+9*Y1^5*Y2^4*X1^6+48*Y1^4*Y2^5*X1^5*X2+70*Y1^3*Y2^6*X1^4*X2^2+48*Y1^2*Y2^7*X1^3*X2^3+13*Y1*Y2^8*X1^2*X2^4+2*Y2^9*X1*X2^5,Y1^7*Y2^4*X1^5*X2+11*Y1^5*Y2^6*X1^3*X2^3-4*Y1^4*Y2^7*X1^2*X2^4-28*Y1^3*Y2^8*X1*X2^5-28*Y1^2*Y2^9*X2^6-8*Y1^6*Y2^4*X1^6-62*Y1^5*Y2^5*X1^5*X2-70*Y1^4*Y2^6*X1^4*X2^2-54*Y1^3*Y2^7*X1^3*X2^3-44*Y1^2*Y2^8*X1^2*X2^4-44*Y1*Y2^9*X1*X2^5-6*Y2^10*X2^6-42*Y1^4*Y2^5*X1^6-67*Y1^3*Y2^6*X1^5*X2-48*Y1^2*Y2^7*X1^4*X2^2-13*Y1*Y2^8*X1^3*X2^3-6*Y2^9*X1^2*X2^4,52*Y1^8*Y2^3*X1^2*X2^4+90*Y1^7*Y2^4*X1*X2^5+60*Y1^6*Y2^5*X2^6+Y1^10*X1^6-16*Y1^9*Y2*X1^5*X2-127*Y1^8*Y2^2*X1^4*X2^2-136*Y1^7*Y2^3*X1^3*X2^3-25*Y1^6*Y2^4*X1^2*X2^4+52*Y1^5*Y2^5*X1*X2^5+7*Y1^4*Y2^6*X2^6-7*Y1^8*Y2*X1^6-112*Y1^7*Y2^2*X1^5*X2-203*Y1^6*Y2^3*X1^4*X2^2-168*Y1^5*Y2^4*X1^3*X2^3-61*Y1^4*Y2^5*X1^2*X2^4-14*Y1^3*Y2^6*X1*X2^5-Y1^2*Y2^7*X2^6,13*Y1^8*Y2^3*X1^3*X2^3-23*Y1^6*Y2^5*X1*X2^5-24*Y1^5*Y2^6*X2^6-3*Y1^9*Y2*X1^6-4*Y1^8*Y2^2*X1^5*X2+95*Y1^7*Y2^3*X1^4*X2^2+174*Y1^6*Y2^4*X1^3*X2^3+127*Y1^5*Y2^5*X1^2*X2^4+26*Y1^4*Y2^6*X1*X2^5+5*Y1^3*Y2^7*X2^6-5*Y1^7*Y2^2*X1^6+76*Y1^6*Y2^3*X1^5*X2+193*Y1^5*Y2^4*X1^4*X2^2+205*Y1^4*Y2^5*X1^3*X2^3+105*Y1^3*Y2^6*X1^2*X2^4+29*Y1^2*Y2^7*X1*X2^5+3*Y1*Y2^8*X2^6,13*Y1^9*Y2^2*X1^3*X2^3-23*Y1^7*Y2^4*X1*X2^5-24*Y1^6*Y2^5*X2^6-3*Y1^10*X1^6-4*Y1^9*Y2*X1^5*X2+95*Y1^8*Y2^2*X1^4*X2^2+174*Y1^7*Y2^3*X1^3*X2^3+127*Y1^6*Y2^4*X1^2*X2^4+26*Y1^5*Y2^5*X1*X2^5+5*Y1^4*Y2^6*X2^6-5*Y1^8*Y2*X1^6+76*Y1^7*Y2^2*X1^5*X2+193*Y1^6*Y2^3*X1^4*X2^2+205*Y1^5*Y2^4*X1^3*X2^3+105*Y1^4*Y2^5*X1^2*X2^4+29*Y1^3*Y2^6*X1*X2^5+3*Y1^2*Y2^7*X2^6,4*Y1^9*Y2*X1^7*X2+28*Y1^8*Y2^2*X1^6*X2^2+84*Y1^7*Y2^3*X1^5*X2^3+140*Y1^6*Y2^4*X1^4*X2^4+140*Y1^5*Y2^5*X1^3*X2^5+84*Y1^4*Y2^6*X1^2*X2^6+28*Y1^3*Y2^7*X1*X2^7+4*Y1^2*Y2^8*X2^8+7*Y1^8*Y2*X1^8+52*Y1^7*Y2^2*X1^7*X2+168*Y1^6*Y2^3*X1^6*X2^2+308*Y1^5*Y2^4*X1^5*X2^3+350*Y1^4*Y2^5*X1^4*X2^4+252*Y1^3*Y2^6*X1^3*X2^5+112*Y1^2*Y2^7*X1^2*X2^6+28*Y1*Y2^8*X1*X2^7+3*Y2^9*X2^8,3*Y1^9*Y2*X1^8+28*Y1^8*Y2^2*X1^7*X2+112*Y1^7*Y2^3*X1^6*X2^2+252*Y1^6*Y2^4*X1^5*X2^3+350*Y1^5*Y2^5*X1^4*X2^4+308*Y1^4*Y2^6*X1^3*X2^5+168*Y1^3*Y2^7*X1^2*X2^6+52*Y1^2*Y2^8*X1*X2^7+7*Y1*Y2^9*X2^8+4*Y1^7*Y2^2*X1^8+28*Y1^6*Y2^3*X1^7*X2+84*Y1^5*Y2^4*X1^6*X2^2+140*Y1^4*Y2^5*X1^5*X2^3+140*Y1^3*Y2^6*X1^4*X2^4+84*Y1^2*Y2^7*X1^3*X2^5+28*Y1*Y2^8*X1^2*X2^6+4*Y2^9*X1*X2^7,Y1^10*X1^8-28*Y1^8*Y2^2*X1^6*X2^2-112*Y1^7*Y2^3*X1^5*X2^3-210*Y1^6*Y2^4*X1^4*X2^4-224*Y1^5*Y2^5*X1^3*X2^5-140*Y1^4*Y2^6*X1^2*X2^6-48*Y1^3*Y2^7*X1*X2^7-7*Y1^2*Y2^8*X2^8-15*Y1^8*Y2*X1^8-112*Y1^7*Y2^2*X1^7*X2-364*Y1^6*Y2^3*X1^6*X2^2-672*Y1^5*Y2^4*X1^5*X2^3-770*Y1^4*Y2^5*X1^4*X2^4-560*Y1^3*Y2^6*X1^3*X2^5-252*Y1^2*Y2^7*X1^2*X2^6-64*Y1*Y2^8*X1*X2^7-7*Y2^9*X2^8,5*Y1^7*Y2^4*X1^7-113*Y1^5*Y2^6*X1^5*X2^2-762*Y1^4*Y2^7*X1^4*X2^3-1196*Y1^3*Y2^8*X1^3*X2^4-800*Y1^2*Y2^9*X1^2*X2^5-80*Y1*Y2^10*X1*X2^6+2*Y2^11*X2^7+106*Y1^5*Y2^5*X1^7+726*Y1^4*Y2^6*X1^6*X2+402*Y1^3*Y2^7*X1^5*X2^2-572*Y1^2*Y2^8*X1^4*X2^3-876*Y1*Y2^9*X1^3*X2^4-170*Y2^10*X1^2*X2^5+545*Y1^3*Y2^6*X1^7+876*Y1^2*Y2^7*X1^6*X2+567*Y1*Y2^8*X1^5*X2^2+60*Y2^9*X1^4*X2^3,Y1^8*Y2^2*X1^8*X2+4*Y1^7*Y2^3*X1^7*X2^2-28*Y1^5*Y2^5*X1^5*X2^4-70*Y1^4*Y2^6*X1^4*X2^5-84*Y1^3*Y2^7*X1^3*X2^6-56*Y1^2*Y2^8*X1^2*X2^7-20*Y1*Y2^9*X1*X2^8-3*Y2^10*X2^9+4*Y1^7*Y2^2*X1^9+28*Y1^6*Y2^3*X1^8*X2+84*Y1^5*Y2^4*X1^7*X2^2+140*Y1^4*Y2^5*X1^6*X2^3+140*Y1^3*Y2^6*X1^5*X2^4+84*Y1^2*Y2^7*X1^4*X2^5+28*Y1*Y2^8*X1^3*X2^6+4*Y2^9*X1^2*X2^7]
#draw gr^i in the shape of a matrix.
Rp = macaulay2('QQ[X1,X2]')
mat = macaulay2(toMacau(arr2,2))
MMM2 = mat.coker()
print("gr²:")
print(MMM2)
mat = macaulay2(toMacau(arr3,3))
MMM3 = mat.coker()
print("gr³:")
print(MMM3)
mat = macaulay2(toMacau(arr4,4))
MMM4 = mat.coker()
print("gr⁴:")
print(MMM4)
mat = macaulay2(toMacau(arr5,5))
MMM5 = mat.coker()
print("gr⁵:")
print(MMM5)
mat = macaulay2(toMacau(arr6,6))
MMM6 = mat.coker()
print("gr⁶:")
print(MMM6)
mat = macaulay2(toMacau(arr7,7))
MMM7 = mat.coker()
print("gr⁷:")
print(MMM7)
mat = macaulay2(toMacau(arr8,8))
MMM8 = mat.coker()
print("gr⁸:")
print(MMM8)
mat = macaulay2(toMacau(arr9,9))
MMM9 = mat.coker()
print("gr⁹:")
print(MMM9)
gr²:
cokernel | 0 0 0 0 0 0 |
| 0 0 0 0 0 0 |
| X1^2 0 0 0 0 0 |
gr³:
cokernel | 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 |
| 0 X1^2 0 0 0 0 0 |
| X1^2 0 0 0 0 0 0 |
gr⁴:
cokernel | 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 |
| 0 0 X1^2 0 0 0 0 0 |
| 0 X1^2 0 0 0 0 0 0 |
| X1^2 0 0 369X1^4 290X1^6 0 0 0 |
gr⁵:
cokernel | 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 |
| 0 0 0 X1^2 0 0 0 0 0 |
| 0 0 X1^2 0 -26X1^4 -8X1^6 0 0 0 |
| 0 X1^2 0 0 -32X1^3X2 -27X1^5X2 -157X1^7 0 0 |
| X1^2 0 0 0 -17X1^2X2^2 -44X1^4X2^2 -273X1^6X2 17X1^6 0 |
gr⁶:
cokernel | 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 X1^2 17X1^4 -2X1^6 0 0 0 |
| 0 0 0 X1^2 0 32X1^3X2 -3X1^5X2 5X1^7 0 0 |
| 0 0 X1^2 0 0 26X1^2X2^2 0 21X1^6X2 -X1^6 0 |
| 0 X1^2 0 0 0 8X1X2^3 3X1^3X2^3 27X1^5X2^2 -2X1^5X2 -6X1^6 |
| X1^2 0 0 0 0 X2^4 2X1^2X2^4 15X1^4X2^3 -6X1^4X2^2 -23X1^5X2 |
gr⁷:
cokernel | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 0 -12X1^6 0 0 |
| 0 0 0 0 0 X1^2 0 17X1^4 -2X1^6 0 0 -36X1^5X2 0 0 |
| 0 0 0 0 X1^2 0 17X1^4 32X1^3X2 -20X1^5X2 5X1^6 0 -10X1^4X2^2 -42X1^6 0 |
| 0 0 0 X1^2 0 0 32X1^3X2 26X1^2X2^2 -32X1^4X2^2 8X1^5X2 -12X1^6 37X1^3X2^3 -67X1^5X2 545X1^7 |
| 0 0 X1^2 0 0 0 26X1^2X2^2 8X1X2^3 -23X1^3X2^3 6X1^4X2^2 -19X1^5X2 44X1^2X2^4 -48X1^4X2^2 876X1^6X2 |
| 0 X1^2 0 0 0 0 8X1X2^3 X2^4 -6X1^2X2^4 2X1^3X2^3 -12X1^4X2^2 15X1X2^5 -13X1^3X2^3 567X1^5X2^2 |
| X1^2 0 0 0 0 0 X2^4 0 -X1X2^5 X1^2X2^4 -X1^3X2^3 2X2^6 -6X1^2X2^4 60X1^4X2^3 |
gr⁸:
cokernel | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -12X1^6 0 12X1^9 |
| 0 0 0 0 0 0 X1^2 0 0 17X1^4 -7X1^6 0 0 0 -5X1^6 0 0 -36X1^5X2 0 84X1^8X2 |
| 0 0 0 0 0 X1^2 0 0 17X1^4 32X1^3X2 -112X1^5X2 -2X1^6 0 0 76X1^5X2 9X1^6 0 -10X1^4X2^2 0 252X1^7X2^2 |
| 0 0 0 0 X1^2 0 0 17X1^4 32X1^3X2 26X1^2X2^2 -203X1^4X2^2 -20X1^5X2 5X1^6 0 193X1^4X2^2 48X1^5X2 -42X1^6 37X1^3X2^3 0 420X1^6X2^3 |
| 0 0 0 X1^2 0 0 0 32X1^3X2 26X1^2X2^2 8X1X2^3 -168X1^3X2^3 -32X1^4X2^2 8X1^5X2 -12X1^6 205X1^3X2^3 70X1^4X2^2 -67X1^5X2 44X1^2X2^4 545X1^7 420X1^5X2^4 |
| 0 0 X1^2 0 0 0 0 26X1^2X2^2 8X1X2^3 X2^4 -61X1^2X2^4 -23X1^3X2^3 6X1^4X2^2 -19X1^5X2 105X1^2X2^4 48X1^3X2^3 -48X1^4X2^2 15X1X2^5 876X1^6X2 252X1^4X2^5 |
| 0 X1^2 0 0 0 0 0 8X1X2^3 X2^4 0 -14X1X2^5 -6X1^2X2^4 2X1^3X2^3 -12X1^4X2^2 29X1X2^5 13X1^2X2^4 -13X1^3X2^3 2X2^6 567X1^5X2^2 84X1^3X2^6 |
| X1^2 0 0 0 0 0 0 X2^4 0 0 -X2^6 -X1X2^5 X1^2X2^4 -X1^3X2^3 3X2^6 2X1X2^5 -6X1^2X2^4 0 60X1^4X2^3 12X1^2X2^7 |
gr⁹:
cokernel | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -7X1^6 0 -5X1^6 7X1^8 0 -15X1^8 0 0 |
| 0 0 0 0 0 0 0 X1^2 0 0 0 17X1^4 0 0 0 0 -7X1^6 0 0 0 -112X1^5X2 -5X1^6 76X1^5X2 52X1^7X2 4X1^8 -112X1^7X2 0 4X1^9 |
| 0 0 0 0 0 0 X1^2 0 0 0 17X1^4 32X1^3X2 -7X1^6 0 0 0 -112X1^5X2 -5X1^6 0 0 -203X1^4X2^2 76X1^5X2 193X1^4X2^2 168X1^6X2^2 28X1^7X2 -364X1^6X2^2 0 28X1^8X2 |
| 0 0 0 0 0 X1^2 0 0 0 17X1^4 32X1^3X2 26X1^2X2^2 -112X1^5X2 -2X1^6 0 0 -203X1^4X2^2 76X1^5X2 9X1^6 0 -168X1^3X2^3 193X1^4X2^2 205X1^3X2^3 308X1^5X2^3 84X1^6X2^2 -672X1^5X2^3 0 84X1^7X2^2 |
| 0 0 0 0 X1^2 0 0 0 17X1^4 32X1^3X2 26X1^2X2^2 8X1X2^3 -203X1^4X2^2 -20X1^5X2 5X1^6 0 -168X1^3X2^3 193X1^4X2^2 48X1^5X2 -42X1^6 -61X1^2X2^4 205X1^3X2^3 105X1^2X2^4 350X1^4X2^4 140X1^5X2^3 -770X1^4X2^4 0 140X1^6X2^3 |
| 0 0 0 X1^2 0 0 0 0 32X1^3X2 26X1^2X2^2 8X1X2^3 X2^4 -168X1^3X2^3 -32X1^4X2^2 8X1^5X2 -12X1^6 -61X1^2X2^4 205X1^3X2^3 70X1^4X2^2 -67X1^5X2 -14X1X2^5 105X1^2X2^4 29X1X2^5 252X1^3X2^5 140X1^4X2^4 -560X1^3X2^5 545X1^7 140X1^5X2^4 |
| 0 0 X1^2 0 0 0 0 0 26X1^2X2^2 8X1X2^3 X2^4 0 -61X1^2X2^4 -23X1^3X2^3 6X1^4X2^2 -19X1^5X2 -14X1X2^5 105X1^2X2^4 48X1^3X2^3 -48X1^4X2^2 -X2^6 29X1X2^5 3X2^6 112X1^2X2^6 84X1^3X2^5 -252X1^2X2^6 876X1^6X2 84X1^4X2^5 |
| 0 X1^2 0 0 0 0 0 0 8X1X2^3 X2^4 0 0 -14X1X2^5 -6X1^2X2^4 2X1^3X2^3 -12X1^4X2^2 -X2^6 29X1X2^5 13X1^2X2^4 -13X1^3X2^3 0 3X2^6 0 28X1X2^7 28X1^2X2^6 -64X1X2^7 567X1^5X2^2 28X1^3X2^6 |
| X1^2 0 0 0 0 0 0 0 X2^4 0 0 0 -X2^6 -X1X2^5 X1^2X2^4 -X1^3X2^3 0 3X2^6 2X1X2^5 -6X1^2X2^4 0 0 0 3X2^8 4X1X2^7 -7X2^8 60X1^4X2^3 4X1^2X2^7 |
It is already clear that $\mathrm{gr}^i = \mathbf{R}Y_1^i \oplus \mathbf{R}Y_1^{i-1}Y_2$ for $1\leq i\leq 6$. For $i=7,8,9$ we need to prune torsion to be sure:
In [52]:
NN = MMM7 / MMM7.torsionSubmodule()
print("gr⁷:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM8 / MMM8.torsionSubmodule()
print("gr⁸:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM9 / MMM9.torsionSubmodule()
print("gr⁹:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
gr⁷:
1
(QQ[X1, X2])
rank = 1
-------------------
gr⁸:
1
(QQ[X1, X2])
rank = 1
-------------------
gr⁹:
1
(QQ[X1, X2])
rank = 1
-------------------
Therefore we have $\mathrm{gr}^i = \mathbf{R}Y_1^i$ for $i=7,8,9$. Summing everything up, we get $m(t) = [10]_{t^w} + t^{2w}[6]_{t^w}$. Moreover, we can only have $w=2$ for this local equation, so that the final answer is $$m(t) = [10]_{t^2} + t^4[6]_{t^2}$$ Note that in this case, all pieces of associated graded are actually locally free, which is a minor miracle!
Case $x^3y^4$¶
Once again, the usual multiplicity is $16$. Since the polynomials $p_1$, $p_2$ have no linear terms, we may start from $\mathrm{gr}^2$.
In [53]:
R = macaulay2('QQ[Y1,Y2,X1,X2,MonomialOrder=>ProductOrder{2,2}]','R')
I = macaulay2('ideal{Y1,Y2}')
P1 = macaulay2('Y2^2*X1^3 + 6*Y1^2*Y2*X1^3 + 12*Y1*Y2^2*X1^2*X2 + 3*Y2^3*X1*X2^2 + Y1^4*X1^3 + 12*Y1^3*Y2*X1^2*X2 + 18*Y1^2*Y2^2*X1*X2^2 + 4*Y1*Y2^3*X2^3 + 3*Y1^4*Y2*X1*X2^2 + 4*Y1^3*Y2^2*X2^3')
P2 = macaulay2('Y2^4*X1^6 - 4*Y1^2*Y2^3*X1^6 - 3*Y2^5*X1^4*X2^2 + 6*Y1^4*Y2^2*X1^6 + 12*Y1^2*Y2^4*X1^4*X2^2 + 3*Y2^6*X1^2*X2^4 - 4*Y1^6*Y2*X1^6 - 18*Y1^4*Y2^3*X1^4*X2^2 - 12*Y1^2*Y2^5*X1^2*X2^4 - Y2^7*X2^6 + Y1^8*X1^6 + 12*Y1^6*Y2^2*X1^4*X2^2 + 18*Y1^4*Y2^4*X1^2*X2^4 + 4*Y1^2*Y2^6*X2^6 - 3*Y1^8*Y2*X1^4*X2^2 - 12*Y1^6*Y2^3*X1^2*X2^4 - 6*Y1^4*Y2^5*X2^6 + 3*Y1^8*Y2^2*X1^2*X2^4 + 4*Y1^6*Y2^4*X2^6 - Y1^8*Y2^3*X2^6')
J34 = macaulay2.ideal(P1,P2)
#print("I∩J:",str(macaulay2.intersect(I,J34).toString())[6:-1],"\n")
print("I²∩J:",str(macaulay2.intersect(I^2,J34).toString())[6:-1],"\n")
print("I³∩J:",str(macaulay2.intersect(I^3,J34).toString())[6:-1],"\n")
print("I⁴∩J:",str(macaulay2.intersect(I^4,J34).toString())[6:-1],"\n")
print("I⁵∩J:",str(macaulay2.intersect(I^5,J34).toString())[6:-1],"\n")
print("I⁶∩J:",str(macaulay2.intersect(I^6,J34).toString())[6:-1],"\n")
print("I⁷∩J:",str(macaulay2.intersect(I^7,J34).toString())[6:-1],"\n")
print("I⁸∩J:",str(macaulay2.intersect(I^8,J34).toString())[6:-1],"\n")
print("I⁹∩J:",str(macaulay2.intersect(I^9,J34).toString())[6:-1],"\n")
I²∩J: 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I³∩J: 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In [54]:
arr2 = 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arr6 = 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6*X2^6-1134*Y1^9*Y2*X1^6-12852*Y1^8*Y2^2*X1^5*X2-28116*Y1^7*Y2^3*X1^4*X2^2-24288*Y1^6*Y2^4*X1^3*X2^3-2766*Y1^5*Y2^5*X1^2*X2^4+420*Y1^4*Y2^6*X1*X2^5+81*Y1^3*Y2^7*X2^6-5373*Y1^7*Y2^2*X1^6-14724*Y1^6*Y2^3*X1^5*X2-16710*Y1^5*Y2^4*X1^4*X2^2-7636*Y1^4*Y2^5*X1^3*X2^3-1701*Y1^3*Y2^6*X1^2*X2^4]
#draw gr^i in the shape of a matrix.
Rp = macaulay2('QQ[X1,X2]')
mat = macaulay2(toMacau(arr2,2))
MMM2 = mat.coker()
print("gr²:")
print(MMM2)
mat = macaulay2(toMacau(arr3,3))
MMM3 = mat.coker()
print("gr³:")
print(MMM3)
mat = macaulay2(toMacau(arr4,4))
MMM4 = mat.coker()
print("gr⁴:")
print(MMM4)
mat = macaulay2(toMacau(arr5,5))
MMM5 = mat.coker()
print("gr⁵:")
print(MMM5)
mat = macaulay2(toMacau(arr6,6))
MMM6 = mat.coker()
print("gr⁶:")
print(MMM6)
mat = macaulay2(toMacau(arr7,7))
MMM7 = mat.coker()
print("gr⁷:")
print(MMM7)
mat = macaulay2(toMacau(arr8,8))
MMM8 = mat.coker()
print("gr⁸:")
print(MMM8)
mat = macaulay2(toMacau(arr9,9))
MMM9 = mat.coker()
print("gr⁹:")
print(MMM9)
gr²:
cokernel | 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 |
| X1^3 0 0 0 0 0 0 0 0 0 |
gr³:
cokernel | 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 0 |
| 0 X1^3 0 0 0 0 0 0 0 0 0 |
| X1^3 0 0 0 0 0 0 0 0 0 0 |
gr⁴:
cokernel | 0 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 X1^3 0 0 0 0 0 0 0 0 0 |
| 0 X1^3 0 0 0 0 0 0 0 0 0 0 |
| X1^3 0 0 -160303X1^6 49392091X1^8 0 0 0 0 0 0 0 |
gr⁵:
cokernel | 0 0 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 X1^3 0 0 0 0 0 0 0 0 0 |
| 0 0 X1^3 0 20322X1^6 -1979208X1^8 0 0 0 0 0 0 0 |
| 0 X1^3 0 0 3804X1^5X2 -1565508X1^7X2 -105714X1^8 0 0 -50176X1^7 0 0 0 |
| X1^3 0 0 0 1903X1^4X2^2 -3553291X1^6X2^2 -1253X1^7X2 -84056X1^8 1810592X1^7 -231X1^6X2 0 -953813X1^9 -878767X1^10 |
gr⁶:
cokernel | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 X1^3 -5373X1^6 99792X1^8 0 0 0 0 0 0 0 |
| 0 0 0 X1^3 0 -14724X1^5X2 325836X1^7X2 4158X1^8 0 0 7392X1^7 0 0 0 |
| 0 0 X1^3 0 0 -16710X1^4X2^2 518616X1^6X2^2 3087X1^7X2 4620X1^8 -328272X1^7 9373X1^6X2 0 36267X1^9 25806X1^10 |
| 0 X1^3 0 0 0 -7636X1^3X2^3 385140X1^5X2^3 7434X1^6X2^2 644X1^7X2 6349X1^6X2 13860X1^5X2^2 -620256X1^7 31556X1^8X2 35548X1^9X2 |
| X1^3 0 0 0 0 -1701X1^2X2^4 173747X1^4X2^4 77X1^5X2^3 6020X1^6X2^2 -388332X1^5X2^2 1477X1^4X2^3 272195X1^6X2 73913X1^7X2^2 73516X1^8X2^2 |
gr⁷:
cokernel | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 -40824X1^8 0 0 0 0 -7776X1^7 0 0 0 0 0 |
| 0 0 0 0 0 X1^3 -219996X1^7X2 0 0 0 -12096X1^7 -34992X1^6X2 0 0 -5373X1^6 0 0 |
| 0 0 0 0 X1^3 0 -495936X1^6X2^2 189X1^7X2 -462X1^8 28728X1^7 -49392X1^6X2 -66240X1^5X2^2 0 -5373X1^6 -14724X1^5X2 -2079X1^9 -351X1^10 |
| 0 0 0 X1^3 0 0 -597492X1^5X2^3 378X1^6X2^2 -504X1^7X2 5040X1^6X2 -82656X1^5X2^2 -66579X1^4X2^3 37296X1^7 -14724X1^5X2 -16710X1^4X2^2 -5292X1^8X2 -2484X1^9X2 |
| 0 0 X1^3 0 0 0 -405384X1^4X2^4 441X1^5X2^3 -1008X1^6X2^2 26208X1^5X2^2 -69783X1^4X2^3 -37372X1^3X2^4 336X1^6X2 -16710X1^4X2^2 -7636X1^3X2^3 -11277X1^7X2^2 -7272X1^8X2^2 |
| 0 X1^3 0 0 0 0 -146692X1^3X2^5 168X1^4X2^4 -252X1^5X2^3 -24423X1^4X2^3 -29932X1^3X2^4 -10935X1^2X2^5 44688X1^5X2^2 -7636X1^3X2^3 -1701X1^2X2^4 -11424X1^6X2^3 -11472X1^7X2^3 |
| X1^3 0 0 0 0 0 -22113X1^2X2^6 91X1^3X2^5 -602X1^4X2^4 32060X1^3X2^4 -5103X1^2X2^5 -1296X1X2^6 -33859X1^4X2^3 -1701X1^2X2^4 0 -9401X1^5X2^4 -10534X1^6X2^4 |
gr⁸:
cokernel | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 -18X1^8 0 0 0 0 0 -7776X1^7 0 0 0 0 |
| 0 0 0 0 0 0 X1^3 -119X1^7X2 14X1^8 4536X1^7 0 -35X1^9 5928X1^7 -34992X1^6X2 0 0 -5373X1^6 -28X1^10 |
| 0 0 0 0 0 X1^3 0 -336X1^6X2^2 84X1^7X2 24192X1^6X2 -1512X1^7 -210X1^8X2 31856X1^6X2 -66240X1^5X2^2 0 -5373X1^6 -14724X1^5X2 -168X1^9X2 |
| 0 0 0 0 X1^3 0 0 -525X1^5X2^3 210X1^6X2^2 53928X1^5X2^2 -2016X1^6X2 -525X1^7X2^2 71448X1^5X2^2 -66579X1^4X2^3 -5373X1^6 -14724X1^5X2 -16710X1^4X2^2 -420X1^8X2^2 |
| 0 0 0 X1^3 0 0 0 -490X1^4X2^4 280X1^5X2^3 64176X1^4X2^3 -4032X1^5X2^2 -700X1^6X2^3 85479X1^4X2^3 -37372X1^3X2^4 -14724X1^5X2 -16710X1^4X2^2 -7636X1^3X2^3 -560X1^7X2^3 |
| 0 0 X1^3 0 0 0 0 -273X1^3X2^5 210X1^4X2^4 42952X1^3X2^4 -1680X1^4X2^3 -525X1^5X2^4 57476X1^3X2^4 -10935X1^2X2^5 -16710X1^4X2^2 -7636X1^3X2^3 -1701X1^2X2^4 -420X1^6X2^4 |
| 0 X1^3 0 0 0 0 0 -84X1^2X2^6 84X1^3X2^5 15309X1^2X2^5 -2296X1^3X2^4 -210X1^4X2^5 20574X1^2X2^5 -1296X1X2^6 -7636X1^3X2^3 -1701X1^2X2^4 0 -168X1^5X2^5 |
| X1^3 0 0 0 0 0 0 -11X1X2^7 14X1^2X2^6 2268X1X2^6 -189X1^2X2^5 -35X1^3X2^6 3060X1X2^6 0 -1701X1^2X2^4 0 0 -28X1^4X2^6 |
gr⁹:
cokernel | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 9X1^9 33X1^8 0 0 -18X1^8 0 0 0 0 0 9X1^10 0 -7776X1^7 0 0 0 0 |
| 0 0 0 0 0 0 0 X1^3 72X1^8X2 216X1^7X2 -18X1^8 0 -119X1^7X2 178X1^8 0 4536X1^7 0 0 72X1^9X2 5928X1^7 -34992X1^6X2 0 0 0 -5373X1^6 |
| 0 0 0 0 0 0 X1^3 0 252X1^7X2^2 602X1^6X2^2 -119X1^7X2 14X1^8 -336X1^6X2^2 1176X1^7X2 4536X1^7 24192X1^6X2 0 611436X1^7 252X1^8X2^2 31856X1^6X2 -66240X1^5X2^2 0 0 -5373X1^6 -14724X1^5X2 |
| 0 0 0 0 0 X1^3 0 0 504X1^6X2^3 924X1^5X2^3 -336X1^6X2^2 84X1^7X2 -525X1^5X2^3 3318X1^6X2^2 24192X1^6X2 53928X1^5X2^2 -1512X1^7 3262168X1^6X2 504X1^7X2^3 71448X1^5X2^2 -66579X1^4X2^3 0 -5373X1^6 -14724X1^5X2 -16710X1^4X2^2 |
| 0 0 0 0 X1^3 0 0 0 630X1^5X2^4 840X1^4X2^4 -525X1^5X2^3 210X1^6X2^2 -490X1^4X2^4 5180X1^5X2^3 53928X1^5X2^2 64176X1^4X2^3 -2016X1^6X2 7272006X1^5X2^2 630X1^6X2^4 85479X1^4X2^3 -37372X1^3X2^4 -5373X1^6 -14724X1^5X2 -16710X1^4X2^2 -7636X1^3X2^3 |
| 0 0 0 X1^3 0 0 0 0 504X1^4X2^5 448X1^3X2^5 -490X1^4X2^4 280X1^5X2^3 -273X1^3X2^5 4830X1^4X2^4 64176X1^4X2^3 42952X1^3X2^4 -4032X1^5X2^2 8654520X1^4X2^3 504X1^5X2^5 57476X1^3X2^4 -10935X1^2X2^5 -14724X1^5X2 -16710X1^4X2^2 -7636X1^3X2^3 -1701X1^2X2^4 |
| 0 0 X1^3 0 0 0 0 0 252X1^3X2^6 126X1^2X2^6 -273X1^3X2^5 210X1^4X2^4 -84X1^2X2^6 2688X1^3X2^5 42952X1^3X2^4 15309X1^2X2^5 -1680X1^4X2^3 5792108X1^3X2^4 252X1^4X2^6 20574X1^2X2^5 -1296X1X2^6 -16710X1^4X2^2 -7636X1^3X2^3 -1701X1^2X2^4 0 |
| 0 X1^3 0 0 0 0 0 0 72X1^2X2^7 12X1X2^7 -84X1^2X2^6 84X1^3X2^5 -11X1X2^7 826X1^2X2^6 15309X1^2X2^5 2268X1X2^6 -2296X1^3X2^4 2064552X1^2X2^5 72X1^3X2^7 3060X1X2^6 0 -7636X1^3X2^3 -1701X1^2X2^4 0 0 |
| X1^3 0 0 0 0 0 0 0 9X1X2^8 -X2^8 -11X1X2^7 14X1^2X2^6 0 108X1X2^7 2268X1X2^6 0 -189X1^2X2^5 305865X1X2^6 9X1^2X2^8 0 0 -1701X1^2X2^4 0 0 0 |
Once again, it is clear that $\mathrm{gr}^i = \mathbf{R}Y_1^i \oplus \mathbf{R}Y_1^{i-1}Y_2$ for $1\leq i\leq 6$. For $i=7,8,9$ we need to prune torsion to be sure. However, this time let us note that if we arrive at locally free sheaf of rank $1$ in $\mathrm{gr}^7$, we will automatically get the same for $i=8,9$, because the number of generators can't grow.
In [55]:
NN = MMM7 / MMM7.torsionSubmodule()
print("gr⁷:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
gr⁷:
1
(QQ[X1, X2])
rank = 1
-------------------
Therefore just as before, we get $m(t) = [10]_{t^w} + t^{2w}[6]_{t^w}$. For this local equation we necessarily have $w=3$, so that $$m(t) = [10]_{t^3} + t^6[6]_{t^3}$$ Miracles abound!
Case $x^4y^5$¶
In this final case, the usual multiplicity is $25$. Since the polynomials $p_1$, $p_2$ have no linear nor quadratic terms, we may start from $\mathrm{gr}^3$.
In [56]:
R = macaulay2('QQ[Y1,Y2,X1,X2,MonomialOrder=>ProductOrder{2,2}]','R')
I = macaulay2('ideal{Y1,Y2}')
P1 = macaulay2('5*Y1*Y2^2*X1^4 + 4*Y2^3*X1^3*X2 + 10*Y1^3*Y2*X1^4 + 40*Y1^2*Y2^2*X1^3*X2 + 30*Y1*Y2^3*X1^2*X2^2 + 4*Y2^4*X1*X2^3 + Y1^5*X1^4 + 20*Y1^4*Y2*X1^3*X2 + 60*Y1^3*Y2^2*X1^2*X2^2 + 40*Y1^2*Y2^3*X1*X2^3 + 5*Y1*Y2^4*X2^4 + 6*Y1^5*Y2*X1^2*X2^2 + 20*Y1^4*Y2^2*X1*X2^3 + 10*Y1^3*Y2^3*X2^4 + Y1^5*Y2^2*X2^4')
P2 = macaulay2('-Y2^5*X1^8 + 5*Y1^2*Y2^4*X1^8 + 4*Y2^6*X1^6*X2^2 - 10*Y1^4*Y2^3*X1^8 - 20*Y1^2*Y2^5*X1^6*X2^2 - 6*Y2^7*X1^4*X2^4 + 10*Y1^6*Y2^2*X1^8 + 40*Y1^4*Y2^4*X1^6*X2^2 + 30*Y1^2*Y2^6*X1^4*X2^4 + 4*Y2^8*X1^2*X2^6 - 5*Y1^8*Y2*X1^8 - 40*Y1^6*Y2^3*X1^6*X2^2 - 60*Y1^4*Y2^5*X1^4*X2^4 - 20*Y1^2*Y2^7*X1^2*X2^6 - Y2^9*X2^8 + Y1^10*X1^8 + 20*Y1^8*Y2^2*X1^6*X2^2 + 60*Y1^6*Y2^4*X1^4*X2^4 + 40*Y1^4*Y2^6*X1^2*X2^6 + 5*Y1^2*Y2^8*X2^8 - 4*Y1^10*Y2*X1^6*X2^2 - 30*Y1^8*Y2^3*X1^4*X2^4 - 40*Y1^6*Y2^5*X1^2*X2^6 - 10*Y1^4*Y2^7*X2^8 + 6*Y1^10*Y2^2*X1^4*X2^4 + 20*Y1^8*Y2^4*X1^2*X2^6 + 10*Y1^6*Y2^6*X2^8 - 4*Y1^10*Y2^3*X1^2*X2^6 - 5*Y1^8*Y2^5*X2^8 + Y1^10*Y2^4*X2^8')
J45 = macaulay2.ideal(P1,P2)
#print("I∩J:",str(macaulay2.intersect(I,J45).toString())[6:-1],"\n")
#print("I²∩J:",str(macaulay2.intersect(I^2,J45).toString())[6:-1],"\n")
print("I³∩J:",str(macaulay2.intersect(I^3,J45).toString())[6:-1],"\n")
print("I⁴∩J:",str(macaulay2.intersect(I^4,J45).toString())[6:-1],"\n")
print("I⁵∩J:",str(macaulay2.intersect(I^5,J45).toString())[6:-1],"\n")
print("I⁶∩J:",str(macaulay2.intersect(I^6,J45).toString())[6:-1],"\n")
print("I⁷∩J:",str(macaulay2.intersect(I^7,J45).toString())[6:-1],"\n")
print("I⁸∩J:",str(macaulay2.intersect(I^8,J45).toString())[6:-1],"\n")
print("I⁹∩J:",str(macaulay2.intersect(I^9,J45).toString())[6:-1],"\n")
print("I¹⁰∩J:",str(macaulay2.intersect(I^10,J45).toString())[6:-1],"\n")
I³∩J: 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I⁴∩J: 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In [57]:
arr3 = 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Rp = macaulay2('QQ[X1,X2]')
mat = macaulay2(toMacau(arr3,3))
MMM3 = mat.coker()
print("gr³:")
print(MMM3)
mat = macaulay2(toMacau(arr4,4))
MMM4 = mat.coker()
print("gr⁴:")
print(MMM4)
mat = macaulay2(toMacau(arr5,5))
MMM5 = mat.coker()
print("gr⁵:")
print(MMM5)
mat = macaulay2(toMacau(arr6,6))
MMM6 = mat.coker()
print("gr⁶:")
print(MMM6)
mat = macaulay2(toMacau(arr7,7))
MMM7 = mat.coker()
print("gr⁷:")
print(MMM7)
mat = macaulay2(toMacau(arr8,8))
MMM8 = mat.coker()
print("gr⁸:")
print(MMM8)
mat = macaulay2(toMacau(arr9,9))
MMM9 = mat.coker()
print("gr⁹:")
print(MMM9)
mat = macaulay2(toMacau(arr10,10))
MMM10 = mat.coker()
print("gr¹⁰:")
print(MMM10)
gr³:
cokernel | 0 0 0 0 0 0 |
| 0 0 0 0 0 0 |
| 5X1^4 0 0 0 0 0 |
| 4X1^3X2 0 0 0 0 0 |
gr⁴:
cokernel | 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 |
| 0 5X1^4 0 0 0 0 0 |
| 5X1^4 4X1^3X2 0 0 0 0 0 |
| 4X1^3X2 0 0 0 0 0 0 |
gr⁵:
cokernel | 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 |
| 0 0 5X1^4 0 0 0 0 0 |
| 0 5X1^4 4X1^3X2 0 0 0 0 0 |
| 5X1^4 4X1^3X2 0 0 0 0 0 0 |
| 4X1^3X2 0 0 X1^8 0 0 0 0 |
gr⁶:
cokernel | 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 5X1^4 0 0 0 0 0 0 |
| 0 0 5X1^4 4X1^3X2 0 0 0 0 0 0 |
| 0 5X1^4 4X1^3X2 0 0 0 0 0 0 0 |
| 5X1^4 4X1^3X2 0 0 0 -15614656X1^8 -4356469120X1^10 0 0 0 |
| 4X1^3X2 0 0 0 X1^8 -12491725X1^7X2 -3485175445X1^9X2 -3665X1^10 0 0 |
gr⁷:
cokernel | 0 0 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 5X1^4 0 0 0 0 0 0 0 0 |
| 0 0 0 5X1^4 4X1^3X2 0 0 0 0 0 0 0 0 |
| 0 0 5X1^4 4X1^3X2 0 3944160X1^9 0 0 0 31901280X1^10 0 0 0 |
| 0 5X1^4 4X1^3X2 0 0 7852320X1^8X2 0 0 -3903664X1^8 82706400X1^9X2 354386080X1^10 0 0 |
| 5X1^4 4X1^3X2 0 0 0 7459200X1^7X2^2 0 -15614656X1^8 -5882400X1^7X2 113524992X1^8X2^2 702315840X1^9X2 553923116024X1^12 0 |
| 4X1^3X2 0 0 0 0 2961285X1^6X2^3 X1^8 -12491725X1^7X2 -2207575X1^6X2^2 54221355X1^7X2^3 335045593X1^8X2^2 443138477885X1^11X2 13699X1^12 |
gr⁸:
cokernel | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 5X1^4 0 0 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 5X1^4 4X1^3X2 -186480X1^10 0 179136X1^9 0 0 0 0 0 765180X1^11 0 0 0 0 |
| 0 0 0 5X1^4 4X1^3X2 0 -571200X1^9X2 0 397320X1^8X2 986040X1^9 0 0 0 0 2598450X1^10X2 0 0 0 0 |
| 0 0 5X1^4 4X1^3X2 0 0 -931560X1^8X2^2 3944160X1^9 432600X1^7X2^2 1963080X1^8X2 0 0 0 -975916X1^8 4516260X1^9X2^2 0 327397980X1^10 1107570000X1^12 0 |
| 0 5X1^4 4X1^3X2 0 0 0 -1008000X1^7X2^3 7852320X1^8X2 312480X1^6X2^3 1864800X1^7X2^2 0 0 -3903664X1^8 -1470600X1^7X2 4869648X1^8X2^3 354386080X1^10 924824580X1^9X2 3271949280X1^11X2 -1833213920X1^12 |
| 5X1^4 4X1^3X2 0 0 0 0 -724080X1^6X2^4 7459200X1^7X2^2 144480X1^5X2^4 1206240X1^6X2^3 0 -15614656X1^8 -5882400X1^7X2 -869400X1^6X2^2 3191700X1^7X2^4 702315840X1^9X2 1380334320X1^8X2^2 5112124704X1^10X2^2 -3639762240X1^11X2 |
| 4X1^3X2 0 0 0 0 0 -238245X1^5X2^5 2961285X1^6X2^3 33045X1^4X2^5 372735X1^5X2^4 X1^8 -12491725X1^7X2 -2207575X1^6X2^2 -254005X1^5X2^3 935520X1^6X2^5 335045593X1^8X2^2 680007525X1^7X2^3 2562728325X1^9X2^3 -1738552955X1^10X2^2 |
gr⁹:
cokernel | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 5X1^4 60X1^11 0 0 -7740X1^10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 5X1^4 4X1^3X2 420X1^10X2 -28800X1^10 0 -30660X1^9X2 2340X1^10 0 0 0 0 44784X1^9 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 5X1^4 4X1^3X2 0 1260X1^9X2^2 -114240X1^9X2 -186480X1^10 -62580X1^8X2^2 11760X1^9X2 342816X1^10 0 179136X1^9 0 99330X1^8X2 -8952048X1^9 0 0 0 0 0 0 -474255X1^12 0 |
| 0 0 0 5X1^4 4X1^3X2 0 0 2100X1^8X2^3 -233520X1^8X2^2 -571200X1^9X2 -79800X1^7X2^3 27510X1^8X2^2 1110270X1^9X2 0 397320X1^8X2 986040X1^9 108150X1^7X2^2 -21199080X1^8X2 0 0 0 0 -975916X1^8 0 -1856400X1^11X2 -1027520X1^12 |
| 0 0 5X1^4 4X1^3X2 0 0 0 2100X1^7X2^4 -298200X1^7X2^3 -931560X1^8X2^2 -65100X1^6X2^4 37800X1^7X2^3 1885548X1^8X2^2 3944160X1^9 432600X1^7X2^2 1963080X1^8X2 78120X1^6X2^3 -24717000X1^7X2^2 0 0 0 -975916X1^8 -1470600X1^7X2 0 -3744240X1^10X2^2 -2448540X1^11X2 |
| 0 5X1^4 4X1^3X2 0 0 0 0 1260X1^6X2^5 -243600X1^6X2^4 -1008000X1^7X2^3 -33180X1^5X2^5 32340X1^6X2^4 2083800X1^7X2^3 7852320X1^8X2 312480X1^6X2^3 1864800X1^7X2^2 36120X1^5X2^4 -18698400X1^6X2^3 0 0 -3903664X1^8 -1470600X1^7X2 -869400X1^6X2^2 354386080X1^10 -4750200X1^9X2^3 -3148752X1^10X2^2 |
| 5X1^4 4X1^3X2 0 0 0 0 0 420X1^5X2^6 -124320X1^5X2^5 -724080X1^6X2^4 -9660X1^4X2^6 17040X1^5X2^5 1514730X1^6X2^4 7459200X1^7X2^2 144480X1^5X2^4 1206240X1^6X2^3 9705X1^4X2^5 -8938680X1^5X2^4 0 -15614656X1^8 -5882400X1^7X2 -869400X1^6X2^2 -340005X1^5X2^3 702315840X1^9X2 -3940230X1^8X2^4 -2990340X1^9X2^3 |
| 4X1^3X2 0 0 0 0 0 0 60X1^4X2^7 -30465X1^4X2^6 -238245X1^5X2^5 -1230X1^3X2^7 4260X1^4X2^6 501120X1^5X2^5 2961285X1^6X2^3 33045X1^4X2^5 372735X1^5X2^4 1155X1^3X2^6 -2089335X1^4X2^5 X1^8 -12491725X1^7X2 -2207575X1^6X2^2 -254005X1^5X2^3 -68800X1^4X2^4 335045593X1^8X2^2 -1424130X1^7X2^5 -1209851X1^8X2^4 |
gr¹⁰:
cokernel | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 0 0 0 -27500X1^11 0 0 0 0 220X1^10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 5X1^4 15X1^10 0 0 -232485X1^10X2 1155X1^11 0 0 0 549X1^9X2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 5X1^4 4X1^3X2 0 -180X1^10 -45432X1^9 -874500X1^9X2^2 9900X1^10X2 60X1^11 0 0 1740X1^8X2^2 0 0 0 0 -698328X1^9 0 0 0 0 0 0 780X1^12 0 -300X1^13 0 |
| 0 0 0 0 0 5X1^4 4X1^3X2 0 -420X1^8X2^2 -1260X1^9X2 -115920X1^8X2 -1921920X1^8X2^3 36960X1^9X2^2 420X1^10X2 -28800X1^10 0 4200X1^7X2^3 0 0 0 44784X1^9 -2170140X1^8X2 0 0 0 0 0 0 5460X1^11X2 0 -2100X1^12X2 0 |
| 0 0 0 0 5X1^4 4X1^3X2 0 0 -1680X1^7X2^3 -3780X1^8X2^2 -147000X1^7X2^2 -2721180X1^7X2^4 78540X1^8X2^3 1260X1^9X2^2 -114240X1^9X2 -186480X1^10 6300X1^6X2^4 0 179136X1^9 0 99330X1^8X2 -3215100X1^7X2^2 0 0 0 0 0 -975916X1^8 16380X1^10X2^2 0 -6300X1^11X2^2 0 |
| 0 0 0 5X1^4 4X1^3X2 0 0 0 -3150X1^6X2^4 -6300X1^7X2^3 -118650X1^6X2^3 -2575650X1^6X2^5 103950X1^7X2^4 2100X1^8X2^3 -233520X1^8X2^2 -571200X1^9X2 6090X1^5X2^5 0 397320X1^8X2 986040X1^9 108150X1^7X2^2 -2833530X1^6X2^3 0 0 0 0 -975916X1^8 -1470600X1^7X2 27300X1^9X2^3 0 -10500X1^10X2^3 -1027520X1^12 |
| 0 0 5X1^4 4X1^3X2 0 0 0 0 -3360X1^5X2^5 -6300X1^6X2^4 -59220X1^5X2^4 -1630860X1^5X2^6 87780X1^6X2^5 2100X1^7X2^4 -298200X1^7X2^3 -931560X1^8X2^2 3780X1^4X2^6 3944160X1^9 432600X1^7X2^2 1963080X1^8X2 78120X1^6X2^3 -1488900X1^5X2^4 0 0 0 -975916X1^8 -1470600X1^7X2 -869400X1^6X2^2 27300X1^8X2^4 0 -10500X1^9X2^4 -2448540X1^11X2 |
| 0 5X1^4 4X1^3X2 0 0 0 0 0 -2100X1^4X2^6 -3780X1^5X2^5 -16440X1^4X2^5 -666600X1^4X2^7 46200X1^5X2^6 1260X1^6X2^5 -243600X1^6X2^4 -1008000X1^7X2^3 1440X1^3X2^7 7852320X1^8X2 312480X1^6X2^3 1864800X1^7X2^2 36120X1^5X2^4 -421710X1^4X2^5 0 0 -3903664X1^8 -1470600X1^7X2 -869400X1^6X2^2 -340005X1^5X2^3 16380X1^7X2^5 354386080X1^10 -6300X1^8X2^5 -3148752X1^10X2^2 |
| 5X1^4 4X1^3X2 0 0 0 0 0 0 -720X1^3X2^7 -1260X1^4X2^6 -1770X1^3X2^6 -159720X1^3X2^8 13860X1^4X2^7 420X1^5X2^6 -124320X1^5X2^5 -724080X1^6X2^4 300X1^2X2^8 7459200X1^7X2^2 144480X1^5X2^4 1206240X1^6X2^3 9705X1^4X2^5 -42540X1^3X2^6 0 -15614656X1^8 -5882400X1^7X2 -869400X1^6X2^2 -340005X1^5X2^3 -79220X1^4X2^4 5460X1^6X2^6 702315840X1^9X2 -2100X1^7X2^6 -2990340X1^9X2^3 |
| 4X1^3X2 0 0 0 0 0 0 0 -105X1^2X2^8 -180X1^3X2^7 75X1^2X2^7 -17105X1^2X2^9 1815X1^3X2^8 60X1^4X2^7 -30465X1^4X2^6 -238245X1^5X2^5 25X1X2^9 2961285X1^6X2^3 33045X1^4X2^5 372735X1^5X2^4 1155X1^3X2^6 3075X1^2X2^7 X1^8 -12491725X1^7X2 -2207575X1^6X2^2 -254005X1^5X2^3 -68800X1^4X2^4 -8336X1^3X2^5 780X1^5X2^7 335045593X1^8X2^2 -300X1^6X2^7 -1209851X1^8X2^4 |
This looks very messy, so let us prune:
In [58]:
NN = MMM3 / MMM3.torsionSubmodule()
print("gr³:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM4 / MMM4.torsionSubmodule()
print("gr⁴:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM5 / MMM5.torsionSubmodule()
print("gr⁵:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM6 / MMM6.torsionSubmodule()
print("gr⁶:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM7 / MMM7.torsionSubmodule()
print("gr⁷:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM8 / MMM8.torsionSubmodule()
print("gr⁸:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM9 / MMM9.torsionSubmodule()
print("gr⁹:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM10 / MMM10.torsionSubmodule()
print("gr¹⁰:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
gr³:
cokernel | 0 |
| 0 |
| 5X1 |
| 4X2 |
rank = 3
-------------------
gr⁴:
cokernel | 0 0 |
| 0 0 |
| 5X1 0 |
| 4X2 5X1 |
| 0 4X2 |
rank = 3
-------------------
gr⁵:
2
(QQ[X1, X2])
rank = 2
-------------------
gr⁶:
2
(QQ[X1, X2])
rank = 2
-------------------
gr⁷:
2
(QQ[X1, X2])
rank = 2
-------------------
gr⁸:
2
(QQ[X1, X2])
rank = 2
-------------------
gr⁹:
2
(QQ[X1, X2])
rank = 2
-------------------
gr¹⁰:
1
(QQ[X1, X2])
rank = 1
-------------------
Once again, it is clear that $\mathrm{gr}^i = \mathbf{R}Y_1^i \oplus \mathbf{R}Y_1^{i-1}Y_2$ for $5\leq i\leq 9$ and $i=2$. Since we have locally free sheaf of rank $1$ in $\mathrm{gr}^{10}$, we will automatically get the same for $i=11,12$, and we stop here by total multiplicity count. It remains to figure out what is happening for $i=3,4$. In both cases we have a direct summand of the form $\mathbf{R}Y_1^i \oplus \mathbf{R}Y_1^{i-1}Y_2$. The complement is a torsion-free sheaf or rank $1$ which is not locally free. Since we essentially produced an explicit resolution, their classes in $K$-theory are $$[\mathrm{gr}^3_{\mathrm{tf}}] = t^{5w} + t^{6w} - t^{4w + 1},\quad [\mathrm{gr}^4_{\mathrm{tf}}] = t^{6w} + t^{7w} + t^{8w} - t^{5w + 1} - t^{6w + 1}$$ Summing everything together, we get $m(t) = [13]_{t^w} + t^{2w}[9]_{t^w} + t^{4w}[5]_{t^w} + t^{6w} - t^{4w + 1}[3]_{t^w}$. Finally, the local equation imposes that $w=2$, and so the final answer is $$m(t) = [13]_{t^2} + t^{4}[9]_{t^2} + t^{8}[5]_{t^2} + t^{12} - t^{9}[3]_{t^2}$$ Note that by parity, there are no cancellations.
Double dual correction¶
We can take advantage of the fact that we work with modules over $\mathbf{R}$, and replace $\mathrm{gr}^i$ with their double duals, which are always locally free, Let us denote the resulting (positive!) polynomial by $m'(t)$. This doesn't change anything for $xy^2$, $x^2y^4$ or $x^3y^4$. For $x^2y^3$, we get $m'(t) = [7]_{t^w} + t^{2w} + t^{5w-1}$. For $x^4y^5$, we get $m'(t) = [13]_{t^2} + t^4[9]_{t^2} + t^8 + t^{13} + t^{18}$. Note that the behaviour depends on parity of the power of $y$ in the local equation!
Homogeneous case¶
In the atlas $\psi_2$, all equations are homogeneous, which simplifies our setup significantly. We know from the get-go that the intersection $I^k\cap J$ is given by monomials in $Y_i$ and $p_i$, so we don't need to intersect ideals in macaulay2; moreover, we don't need to truncate anything. Instead we have $$\mathrm{gr}^d = \Big(\bigoplus_{i+j=d} \mathbf{R} Y_1^iY_2^j\Big)/\Big(\bigoplus_{i+j=d-\deg_Y p_1} \mathbf{R} Y_1^iY_2^j p_1 \oplus \bigoplus_{i+j=d-\deg_Y p_2} \mathbf{R} Y_1^iY_2^j p_2 \Big)$$ On the other hand, non-equivariant multiplicity is $2b^2$, so necessarily our computations have to become bigger. Recall that in the homogeneous case, we have $\mathrm{wt}(Y_1) = w$, $\mathrm{wt}(Y_2) = w+l = 2w-1$; $\mathrm{wt}(X_1) = -l = -w+1$, $\mathrm{wt}(X_2) = -2l = -2w+2$.
Case $xy^2$¶
The usual multiplicity is $8$. We have $\deg_Y p_1 = 2$, $\deg_Y p_2 = 4$, therefore we may start from $\mathrm{gr}^2$. Furthermore, simply by looking at ranks of modules and comparing with non-equivariant multiplicity, we only need to go until $\mathrm{gr}^4$.
In [59]:
P1 = Y1^2*X1 + 2*Y1*Y2*X2 + Y2^2*X1*X2
P2 = -Y1^4*X2 + Y1^4*X1^2 + 2*Y1^2*Y2^2*X2^2 - 2*Y1^2*Y2^2*X1^2*X2 - Y2^4*X2^3 + Y2^4*X1^2*X2^2
arr2 = [P1]
arr3 = [Y1*P1,Y2*P1]
arr4 = [Y1^2*P1,Y1*Y2*P1,Y2^2*P1,P2]
Rp = macaulay2('QQ[X1,X2]')
mat = macaulay2(toMacau(arr2,2))
MMM2 = mat.coker()
print("gr²:")
print(MMM2)
mat = macaulay2(toMacau(arr3,3))
MMM3 = mat.coker()
print("gr³:")
print(MMM3)
mat = macaulay2(toMacau(arr4,4))
MMM4 = mat.coker()
print("gr⁴:")
print(MMM4)
gr²:
cokernel | X1 |
| 2X2 |
| X1X2 |
gr³:
cokernel | X1 0 |
| 2X2 X1 |
| X1X2 2X2 |
| 0 X1X2 |
gr⁴:
cokernel | X1 0 0 X1^2-X2 |
| 2X2 X1 0 0 |
| X1X2 2X2 X1 -2X1^2X2+2X2^2 |
| 0 X1X2 2X2 0 |
| 0 0 X1X2 X1^2X2^2-X2^3 |
In [60]:
NN = MMM2 / MMM2.torsionSubmodule()
print("gr²:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM3 / MMM3.torsionSubmodule()
print("gr³:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM4 / MMM4.torsionSubmodule()
print("gr⁴:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
gr²:
cokernel | X1 |
| 2X2 |
| X1X2 |
rank = 2
-------------------
gr³:
cokernel | X1 0 |
| 2X2 X1 |
| X1X2 2X2 |
| 0 X1X2 |
rank = 2
-------------------
gr⁴:
1
(QQ[X1, X2])
rank = 1
-------------------
Note that $\mathrm{gr}^i = \mathrm{gr}^i_{\mathrm{tf}}$ for $i\leq 3$, so that we know the class in $K$-theory already. For $\mathrm{gr}^4$ this is a non-trivial computation, but fortunately we have already performed it in the paper, with the result $[\mathrm{gr}^4_{\mathrm{tf}}] = t^{8w-4}$. Summing everything up, we have $$m(t) = 1 + (t^w + t^{2w-1}) + (t^{2w} + t^{3w-1} - t^{w+1}) + (t^{3w} + t^{4w-1} + t^{5w-2} - t^{2w+1} - t^{3w}) + t^{8w-4}$$ or, written more compactly, $$m(t) = [5]_{t^{2w-1}} + t^{w}[3]_{t^{2w-1}} + t^{2w}[2]_{t^{2w-1}} - t^{w+1}[2]_{t^{w}}$$
Case $x^2y^3$¶
The usual multiplicity is $18$. We have $\deg_Y p_1 = 3$, $\deg_Y p_2 = 6$, therefore we may start from $\mathrm{gr}^3$. Furthermore, simply by looking at ranks of modules and comparing with non-equivariant multiplicity, we only need to go until $\mathrm{gr}^7$.
In [61]:
P1 = Y1^3*X2 + Y1^3*X1^2 + 6*Y1^2*Y2*X1*X2 + 3*Y1*Y2^2*X2^2 + 3*Y1*Y2^2*X1^2*X2 + 2*Y2^3*X1*X2^2
P2 = Y1^6*X2^2 - 2*Y1^6*X1^2*X2 + Y1^6*X1^4 - 3*Y1^4*Y2^2*X2^3 + 6*Y1^4*Y2^2*X1^2*X2^2 - 3*Y1^4*Y2^2*X1^4*X2 + 3*Y1^2*Y2^4*X2^4 - 6*Y1^2*Y2^4*X1^2*X2^3 + 3*Y1^2*Y2^4*X1^4*X2^2 - Y2^6*X2^5 + 2*Y2^6*X1^2*X2^4 - Y2^6*X1^4*X2^3
arr3 = [P1]
arr4 = [Y1*P1,Y2*P1]
arr5 = [Y1^2*P1,Y1*Y2*P1,Y2^2*P1]
arr6 = [Y1^3*P1,Y1^2*Y2*P1,Y1*Y2^2*P1,Y2^3*P1,P2]
arr7 = [Y1^4*P1,Y1^3*Y2*P1,Y1^2*Y2^2*P1,Y1*Y2^3*P1,Y2^4*P1,Y1*P2,Y2*P2]
Rp = macaulay2('QQ[X1,X2]')
mat = macaulay2(toMacau(arr3,3))
MMM3 = mat.coker()
print("gr³:")
print(MMM3)
mat = macaulay2(toMacau(arr4,4))
MMM4 = mat.coker()
print("gr⁴:")
print(MMM4)
mat = macaulay2(toMacau(arr5,5))
MMM5 = mat.coker()
print("gr⁵:")
print(MMM5)
mat = macaulay2(toMacau(arr6,6))
MMM6 = mat.coker()
print("gr⁶:")
print(MMM6)
mat = macaulay2(toMacau(arr7,7))
MMM7 = mat.coker()
print("gr⁷:")
print(MMM7)
gr³:
cokernel | X1^2+X2 |
| 6X1X2 |
| 3X1^2X2+3X2^2 |
| 2X1X2^2 |
gr⁴:
cokernel | X1^2+X2 0 |
| 6X1X2 X1^2+X2 |
| 3X1^2X2+3X2^2 6X1X2 |
| 2X1X2^2 3X1^2X2+3X2^2 |
| 0 2X1X2^2 |
gr⁵:
cokernel | X1^2+X2 0 0 |
| 6X1X2 X1^2+X2 0 |
| 3X1^2X2+3X2^2 6X1X2 X1^2+X2 |
| 2X1X2^2 3X1^2X2+3X2^2 6X1X2 |
| 0 2X1X2^2 3X1^2X2+3X2^2 |
| 0 0 2X1X2^2 |
gr⁶:
cokernel | X1^2+X2 0 0 0 X1^4-2X1^2X2+X2^2 |
| 6X1X2 X1^2+X2 0 0 0 |
| 3X1^2X2+3X2^2 6X1X2 X1^2+X2 0 -3X1^4X2+6X1^2X2^2-3X2^3 |
| 2X1X2^2 3X1^2X2+3X2^2 6X1X2 X1^2+X2 0 |
| 0 2X1X2^2 3X1^2X2+3X2^2 6X1X2 3X1^4X2^2-6X1^2X2^3+3X2^4 |
| 0 0 2X1X2^2 3X1^2X2+3X2^2 0 |
| 0 0 0 2X1X2^2 -X1^4X2^3+2X1^2X2^4-X2^5 |
gr⁷:
cokernel | X1^2+X2 0 0 0 0 X1^4-2X1^2X2+X2^2 0 |
| 6X1X2 X1^2+X2 0 0 0 0 X1^4-2X1^2X2+X2^2 |
| 3X1^2X2+3X2^2 6X1X2 X1^2+X2 0 0 -3X1^4X2+6X1^2X2^2-3X2^3 0 |
| 2X1X2^2 3X1^2X2+3X2^2 6X1X2 X1^2+X2 0 0 -3X1^4X2+6X1^2X2^2-3X2^3 |
| 0 2X1X2^2 3X1^2X2+3X2^2 6X1X2 X1^2+X2 3X1^4X2^2-6X1^2X2^3+3X2^4 0 |
| 0 0 2X1X2^2 3X1^2X2+3X2^2 6X1X2 0 3X1^4X2^2-6X1^2X2^3+3X2^4 |
| 0 0 0 2X1X2^2 3X1^2X2+3X2^2 -X1^4X2^3+2X1^2X2^4-X2^5 0 |
| 0 0 0 0 2X1X2^2 0 -X1^4X2^3+2X1^2X2^4-X2^5 |
In [62]:
NN = MMM3 / MMM3.torsionSubmodule()
print("gr³:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM4 / MMM4.torsionSubmodule()
print("gr⁴:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM5 / MMM5.torsionSubmodule()
print("gr⁵:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM6 / MMM6.torsionSubmodule()
print("gr⁶:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM7 / MMM7.torsionSubmodule()
print("gr⁷:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
gr³:
cokernel | X1^2+X2 |
| 6X1X2 |
| 3X1^2X2+3X2^2 |
| 2X1X2^2 |
rank = 3
-------------------
gr⁴:
cokernel | X1^2+X2 0 |
| 6X1X2 X1^2+X2 |
| 3X1^2X2+3X2^2 6X1X2 |
| 2X1X2^2 3X1^2X2+3X2^2 |
| 0 2X1X2^2 |
rank = 3
-------------------
gr⁵:
cokernel | X1^2+X2 0 0 |
| 6X1X2 X1^2+X2 0 |
| 3X1^2X2+3X2^2 6X1X2 X1^2+X2 |
| 2X1X2^2 3X1^2X2+3X2^2 6X1X2 |
| 0 2X1X2^2 3X1^2X2+3X2^2 |
| 0 0 2X1X2^2 |
rank = 3
-------------------
gr⁶:
cokernel | -18X1X2^2 -1/3X1^2X2-1/3X2^2 6X1X2 X1^2+X2 -54X1 |
| -7X1^2X2^2-7X2^3 -2X1X2^2 7/3X1^2X2+7/3X2^2 6X1X2 -21X1^2-21X2 |
| -6X1X2^3 -X1^2X2^2-X2^3 2X1X2^2 3X1^2X2+3X2^2 -18X1X2 |
| 1/3X1^2X2^3+1/3X2^4 -2/3X1X2^3 -1/9X1^2X2^2-1/9X2^3 2X1X2^2 X1^2X2+X2^2 |
rank = 2
-------------------
gr⁷:
cokernel | -1/3X1^2X2-1/3X2^2 X1^2+X2 8/3X1^2X2^3+8/3X2^4 0 -8/9X1^2X2^2-8/9X2^3 0 7X1^4X2^3-7X2^5 7X1^5X2^2-7X1X2^4 |
| -2/7X1X2^2 6/7X1X2 16/7X1X2^4 0 -16/21X1X2^3 0 6X1^3X2^4-6X1X2^5 6X1^4X2^3-6X1^2X2^4 |
rank = 1
-------------------
Note that $\mathrm{gr}^i = \mathrm{gr}^i_{\mathrm{tf}}$ for $i\leq 5$, so that we know the class in $K$-theory already: $$\sum_{i=0}^5 [\mathrm{gr}^i_{\mathrm{tf}}] = 1 + t^w[2]_{t^{w-1}} + t^{2w}[3]_{t^{w-1}} + (t^{3w}[4]_{t^{w-1}} - t^{w+2}) + (t^{4w}[5]_{t^{w-1}} - t^{2w+2} - t^{3w+1}) + (t^{5w}[6]_{t^{w-1}} - t^{3w+2} - t^{4w+1} - t^{5w})$$ For $i=6,7$, in order to compute the $K$-theory class we need to perform more subtle analysis.
The class of $\mathrm{gr}^6$¶
Let us have a closer look at the torsion submodule of $\mathrm{gr}^6$:
In [63]:
MMM6.torsionSubmodule()
Out[63]:
subquotient (| 0 0 0 0
| 0 0 336 0
| 0 63 0 0
| -54X1 -108X1 932X1^2+1134X2 -101X1^2+7X2
| -21X1^2-21X2 -42X1^2 357X1^3+441X1X2 -42X1^3
| -18X1X2 -36X1X2 348X1^2X2+378X2^2 -15X1^2X2+21X2^2
| X1^2X2+X2^2 2X1^2X2+9X2^2 -17X1^3X2+11X1X2^2 2X1^3X2+16X1X2^2
--------------------------------------------------------------------------------
336 |, | X1^2+X2 0 0
0 | | 6X1X2 X1^2+X2 0
0 | | 3X1^2X2+3X2^2 6X1X2 X1^2+X2
909X1^3-4383X1X2 | | 2X1X2^2 3X1^2X2+3X2^2 6X1X2
378X1^4-1680X1^2X2 | | 0 2X1X2^2 3X1^2X2+3X2^2
135X1^3X2-1629X1X2^2 | | 0 0 2X1X2^2
-18X1^4X2-64X1^2X2^2-144X2^3 | | 0 0 0
--------------------------------------------------------------------------------
0 X1^4-2X1^2X2+X2^2 |)
0 0 |
0 -3X1^4X2+6X1^2X2^2-3X2^3 |
X1^2+X2 0 |
6X1X2 3X1^4X2^2-6X1^2X2^3+3X2^4 |
3X1^2X2+3X2^2 0 |
2X1X2^2 -X1^4X2^3+2X1^2X2^4-X2^5 |
We see that the first three vectors of the monomial basis of $I^6/I^7$ can be expressed in other basis elements up to torsion. After doing this, we have only columns $1$ and $4$ left. Let us check that the resulting module is torsion-free:
In [64]:
temp = macaulay2('matrix{{-54*X1,-101*X1^2+7*X2},{-21*X1^2-21*X2,-42*X1^3},{-18*X1*X2,-15*X1^2*X2+21*X2^2},{X1^2*X2+X2^2,2*X1^3*X2+16*X1*X2^2}}')
modu = temp.coker()
print(modu)
print("Torsion submodule:")
print(modu.torsionSubmodule().prune())
print("Rank:",modu.rank())
cokernel | -54X1 -101X1^2+7X2 |
| -21X1^2-21X2 -42X1^3 |
| -18X1X2 -15X1^2X2+21X2^2 |
| X1^2X2+X2^2 2X1^3X2+16X1X2^2 |
Torsion submodule:
0
Rank: 2
We obtained the expression of our rank $2$ torsion free sheaf as a quotient of rank $4$ free sheaf by rank $2$ free sheaf. The weights can be read off by looking at the first non-trivial row, so that $$[\mathrm{gr}^6_{\mathrm{tf}}] = t^{9w-3}[4]_{t^{w-1}} - t^{7w-1}[2]_{t^{w-1}}$$
The class of $\mathrm{gr}^7$¶
Let us have a closer look at the torsion submodule of $\mathrm{gr}^7$:
In [65]:
MMM7.torsionSubmodule()
Out[65]:
subquotient (| 0 0 0 0 0 3
| 0 0 0 0 0 0
| 0 0 0 63 0 0
| 0 0 0 0 42 0
| 0 21 0 0 0 0
| 126 0 0 0 0 0
| 0 -7X1^2 7X1^2+7X2 7X1^2X2 7X1^3+7X1X2 7X1^2X2^2
| -6X2 -6X1X2 6X1X2 6X1X2^2 6X1^2X2+6X2^2 6X1X2^3
--------------------------------------------------------------------------------
0 |, | X1^2+X2 0 0 0
14 | | 6X1X2 X1^2+X2 0 0
0 | | 3X1^2X2+3X2^2 6X1X2 X1^2+X2 0
0 | | 2X1X2^2 3X1^2X2+3X2^2 6X1X2 X1^2+X2
0 | | 0 2X1X2^2 3X1^2X2+3X2^2 6X1X2
0 | | 0 0 2X1X2^2 3X1^2X2+3X2^2
-7X1^3X2-7X1X2^2 | | 0 0 0 2X1X2^2
-6X1^2X2^2-6X2^3 | | 0 0 0 0
--------------------------------------------------------------------------------
0 X1^4-2X1^2X2+X2^2 0 |)
0 0 X1^4-2X1^2X2+X2^2 |
0 -3X1^4X2+6X1^2X2^2-3X2^3 0 |
0 0 -3X1^4X2+6X1^2X2^2-3X2^3 |
X1^2+X2 3X1^4X2^2-6X1^2X2^3+3X2^4 0 |
6X1X2 0 3X1^4X2^2-6X1^2X2^3+3X2^4 |
3X1^2X2+3X2^2 -X1^4X2^3+2X1^2X2^4-X2^5 0 |
2X1X2^2 0 -X1^4X2^3+2X1^2X2^4-X2^5 |
The first $6$ vectors of the monomial basis of $I^7/I^8$ can be expressed in other basis elements up to torsion. After doing this, we have only column $3$ left. We obtained the expression of our rank $1$ torsion free sheaf as a quotient of rank $2$ free sheaf by rank $1$ free sheaf. The weight of the latter can be read off by looking at the first non-trivial row, so that $$[\mathrm{gr}^7_{\mathrm{tf}}] = t^{13w-6}[2]_{t^{w-1}} - t^{11w-4}$$
Summarizing everything, we get $$m(t) = [8]_{t^{2w-1}} + t^w[7]_{t^{2w-1}} + t^{2w}[5]_{t^{2w-1}} + t^{3w}[4]_{t^{2w-1}} + t^{4w}[2]_{t^{2w-1}} - t^{w+2}[3]_{t^{w}} - t^{3w+1}[2]_{t^{w}} - t^{7w-1} - t^{8w-2} - t^{11w-4}$$
Case $x^2y^4$¶
The usual multiplicity is $32$. We have $\deg_Y p_1 = 4$, $\deg_Y p_2 = 8$, therefore we may start from $\mathrm{gr}^4$. Furthermore, simply by looking at ranks of modules and comparing with non-equivariant multiplicity, we only need to go until $\mathrm{gr}^10$.
In [66]:
P1 = Y1^4*X2 + Y1^4*X1^2 + 8*Y1^3*Y2*X1*X2 + 6*Y1^2*Y2^2*X2^2 + 6*Y1^2*Y2^2*X1^2*X2 + 8*Y1*Y2^3*X1*X2^2 + Y2^4*X2^3 + Y2^4*X1^2*X2^2
P2 = Y1^8*X2^2 - 2*Y1^8*X1^2*X2 + Y1^8*X1^4 - 4*Y1^6*Y2^2*X2^3 + 8*Y1^6*Y2^2*X1^2*X2^2 - 4*Y1^6*Y2^2*X1^4*X2 + 6*Y1^4*Y2^4*X2^4 - 12*Y1^4*Y2^4*X1^2*X2^3 + 6*Y1^4*Y2^4*X1^4*X2^2 - 4*Y1^2*Y2^6*X2^5 + 8*Y1^2*Y2^6*X1^2*X2^4 - 4*Y1^2*Y2^6*X1^4*X2^3 + Y2^8*X2^6 - 2*Y2^8*X1^2*X2^5 + Y2^8*X1^4*X2^4
arr4 = [P1]
arr5 = [Y1*P1,Y2*P1]
arr6 = [Y1^2*P1,Y1*Y2*P1,Y2^2*P1]
arr7 = [Y1^3*P1,Y1^2*Y2*P1,Y1*Y2^2*P1,Y2^3*P1]
arr8 = [Y1^4*P1,Y1^3*Y2*P1,Y1^2*Y2^2*P1,Y1*Y2^3*P1,Y2^4*P1,P2]
arr9 = [Y1^5*P1,Y1^4*Y2*P1,Y1^3*Y2^2*P1,Y1^2*Y2^3*P1,Y1*Y2^4*P1,Y2^5*P1,Y1*P2,Y2*P2]
arr10 = [Y1^6*P1,Y1^5*Y2*P1,Y1^4*Y2^2*P1,Y1^3*Y2^3*P1,Y1^2*Y2^4*P1,Y1*Y2^5*P1,Y2^6*P1,Y1^2*P2,Y1*Y2*P2,Y2^2*P2]
Rp = macaulay2('QQ[X1,X2]')
mat = macaulay2(toMacau(arr4,4))
MMM4 = mat.coker()
print("gr⁴:")
print(MMM4)
mat = macaulay2(toMacau(arr5,5))
MMM5 = mat.coker()
print("gr⁵:")
print(MMM5)
mat = macaulay2(toMacau(arr6,6))
MMM6 = mat.coker()
print("gr⁶:")
print(MMM6)
mat = macaulay2(toMacau(arr7,7))
MMM7 = mat.coker()
print("gr⁷:")
print(MMM7)
mat = macaulay2(toMacau(arr8,8))
MMM8 = mat.coker()
print("gr⁸:")
print(MMM8)
mat = macaulay2(toMacau(arr9,9))
MMM9 = mat.coker()
print("gr⁹:")
print(MMM9)
mat = macaulay2(toMacau(arr10,10))
MMM10 = mat.coker()
print("gr¹⁰:")
print(MMM10)
gr⁴:
cokernel | X1^2+X2 |
| 8X1X2 |
| 6X1^2X2+6X2^2 |
| 8X1X2^2 |
| X1^2X2^2+X2^3 |
gr⁵:
cokernel | X1^2+X2 0 |
| 8X1X2 X1^2+X2 |
| 6X1^2X2+6X2^2 8X1X2 |
| 8X1X2^2 6X1^2X2+6X2^2 |
| X1^2X2^2+X2^3 8X1X2^2 |
| 0 X1^2X2^2+X2^3 |
gr⁶:
cokernel | X1^2+X2 0 0 |
| 8X1X2 X1^2+X2 0 |
| 6X1^2X2+6X2^2 8X1X2 X1^2+X2 |
| 8X1X2^2 6X1^2X2+6X2^2 8X1X2 |
| X1^2X2^2+X2^3 8X1X2^2 6X1^2X2+6X2^2 |
| 0 X1^2X2^2+X2^3 8X1X2^2 |
| 0 0 X1^2X2^2+X2^3 |
gr⁷:
cokernel | X1^2+X2 0 0 0 |
| 8X1X2 X1^2+X2 0 0 |
| 6X1^2X2+6X2^2 8X1X2 X1^2+X2 0 |
| 8X1X2^2 6X1^2X2+6X2^2 8X1X2 X1^2+X2 |
| X1^2X2^2+X2^3 8X1X2^2 6X1^2X2+6X2^2 8X1X2 |
| 0 X1^2X2^2+X2^3 8X1X2^2 6X1^2X2+6X2^2 |
| 0 0 X1^2X2^2+X2^3 8X1X2^2 |
| 0 0 0 X1^2X2^2+X2^3 |
gr⁸:
cokernel | X1^2+X2 0 0 0 0 X1^4-2X1^2X2+X2^2 |
| 8X1X2 X1^2+X2 0 0 0 0 |
| 6X1^2X2+6X2^2 8X1X2 X1^2+X2 0 0 -4X1^4X2+8X1^2X2^2-4X2^3 |
| 8X1X2^2 6X1^2X2+6X2^2 8X1X2 X1^2+X2 0 0 |
| X1^2X2^2+X2^3 8X1X2^2 6X1^2X2+6X2^2 8X1X2 X1^2+X2 6X1^4X2^2-12X1^2X2^3+6X2^4 |
| 0 X1^2X2^2+X2^3 8X1X2^2 6X1^2X2+6X2^2 8X1X2 0 |
| 0 0 X1^2X2^2+X2^3 8X1X2^2 6X1^2X2+6X2^2 -4X1^4X2^3+8X1^2X2^4-4X2^5 |
| 0 0 0 X1^2X2^2+X2^3 8X1X2^2 0 |
| 0 0 0 0 X1^2X2^2+X2^3 X1^4X2^4-2X1^2X2^5+X2^6 |
gr⁹:
cokernel | X1^2+X2 0 0 0 0 0 X1^4-2X1^2X2+X2^2 0 |
| 8X1X2 X1^2+X2 0 0 0 0 0 X1^4-2X1^2X2+X2^2 |
| 6X1^2X2+6X2^2 8X1X2 X1^2+X2 0 0 0 -4X1^4X2+8X1^2X2^2-4X2^3 0 |
| 8X1X2^2 6X1^2X2+6X2^2 8X1X2 X1^2+X2 0 0 0 -4X1^4X2+8X1^2X2^2-4X2^3 |
| X1^2X2^2+X2^3 8X1X2^2 6X1^2X2+6X2^2 8X1X2 X1^2+X2 0 6X1^4X2^2-12X1^2X2^3+6X2^4 0 |
| 0 X1^2X2^2+X2^3 8X1X2^2 6X1^2X2+6X2^2 8X1X2 X1^2+X2 0 6X1^4X2^2-12X1^2X2^3+6X2^4 |
| 0 0 X1^2X2^2+X2^3 8X1X2^2 6X1^2X2+6X2^2 8X1X2 -4X1^4X2^3+8X1^2X2^4-4X2^5 0 |
| 0 0 0 X1^2X2^2+X2^3 8X1X2^2 6X1^2X2+6X2^2 0 -4X1^4X2^3+8X1^2X2^4-4X2^5 |
| 0 0 0 0 X1^2X2^2+X2^3 8X1X2^2 X1^4X2^4-2X1^2X2^5+X2^6 0 |
| 0 0 0 0 0 X1^2X2^2+X2^3 0 X1^4X2^4-2X1^2X2^5+X2^6 |
gr¹⁰:
cokernel | X1^2+X2 0 0 0 0 0 0 X1^4-2X1^2X2+X2^2 0 0 |
| 8X1X2 X1^2+X2 0 0 0 0 0 0 X1^4-2X1^2X2+X2^2 0 |
| 6X1^2X2+6X2^2 8X1X2 X1^2+X2 0 0 0 0 -4X1^4X2+8X1^2X2^2-4X2^3 0 X1^4-2X1^2X2+X2^2 |
| 8X1X2^2 6X1^2X2+6X2^2 8X1X2 X1^2+X2 0 0 0 0 -4X1^4X2+8X1^2X2^2-4X2^3 0 |
| X1^2X2^2+X2^3 8X1X2^2 6X1^2X2+6X2^2 8X1X2 X1^2+X2 0 0 6X1^4X2^2-12X1^2X2^3+6X2^4 0 -4X1^4X2+8X1^2X2^2-4X2^3 |
| 0 X1^2X2^2+X2^3 8X1X2^2 6X1^2X2+6X2^2 8X1X2 X1^2+X2 0 0 6X1^4X2^2-12X1^2X2^3+6X2^4 0 |
| 0 0 X1^2X2^2+X2^3 8X1X2^2 6X1^2X2+6X2^2 8X1X2 X1^2+X2 -4X1^4X2^3+8X1^2X2^4-4X2^5 0 6X1^4X2^2-12X1^2X2^3+6X2^4 |
| 0 0 0 X1^2X2^2+X2^3 8X1X2^2 6X1^2X2+6X2^2 8X1X2 0 -4X1^4X2^3+8X1^2X2^4-4X2^5 0 |
| 0 0 0 0 X1^2X2^2+X2^3 8X1X2^2 6X1^2X2+6X2^2 X1^4X2^4-2X1^2X2^5+X2^6 0 -4X1^4X2^3+8X1^2X2^4-4X2^5 |
| 0 0 0 0 0 X1^2X2^2+X2^3 8X1X2^2 0 X1^4X2^4-2X1^2X2^5+X2^6 0 |
| 0 0 0 0 0 0 X1^2X2^2+X2^3 0 0 X1^4X2^4-2X1^2X2^5+X2^6 |
In [67]:
NN = MMM4 / MMM4.torsionSubmodule()
print("gr⁴:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM5 / MMM5.torsionSubmodule()
print("gr⁵:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM6 / MMM6.torsionSubmodule()
print("gr⁶:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM7 / MMM7.torsionSubmodule()
print("gr⁷:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM8 / MMM8.torsionSubmodule()
print("gr⁸:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM9 / MMM9.torsionSubmodule()
print("gr⁹:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM10 / MMM10.torsionSubmodule()
print("gr¹⁰:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
gr⁴:
cokernel | X1^2+X2 |
| 8X1X2 |
| 6X1^2X2+6X2^2 |
| 8X1X2^2 |
| X1^2X2^2+X2^3 |
rank = 4
-------------------
gr⁵:
cokernel | X1^2+X2 0 |
| 8X1X2 X1^2+X2 |
| 6X1^2X2+6X2^2 8X1X2 |
| 8X1X2^2 6X1^2X2+6X2^2 |
| X1^2X2^2+X2^3 8X1X2^2 |
| 0 X1^2X2^2+X2^3 |
rank = 4
-------------------
gr⁶:
cokernel | X1^2+X2 0 0 |
| 8X1X2 X1^2+X2 0 |
| 6X1^2X2+6X2^2 8X1X2 X1^2+X2 |
| 8X1X2^2 6X1^2X2+6X2^2 8X1X2 |
| X1^2X2^2+X2^3 8X1X2^2 6X1^2X2+6X2^2 |
| 0 X1^2X2^2+X2^3 8X1X2^2 |
| 0 0 X1^2X2^2+X2^3 |
rank = 4
-------------------
gr⁷:
cokernel | X1^2+X2 0 0 0 |
| 8X1X2 X1^2+X2 0 0 |
| 6X1^2X2+6X2^2 8X1X2 X1^2+X2 0 |
| 8X1X2^2 6X1^2X2+6X2^2 8X1X2 X1^2+X2 |
| X1^2X2^2+X2^3 8X1X2^2 6X1^2X2+6X2^2 8X1X2 |
| 0 X1^2X2^2+X2^3 8X1X2^2 6X1^2X2+6X2^2 |
| 0 0 X1^2X2^2+X2^3 8X1X2^2 |
| 0 0 0 X1^2X2^2+X2^3 |
rank = 4
-------------------
gr⁸:
cokernel | -15X1^2+X2 359X1^2X2^2-25X2^3 8X1^3X2 -60X1^2X2+4X2^2 -8X1^3 X1^2+X2 390X1^4X2^2+36492X1^2X2^3-2538X2^4 |
| -18X1^3-10X1X2 432X1^3X2^2+232X1X2^3 9X1^4X2+9X1^2X2^2 -72X1^3X2-40X1X2^2 -9X1^4-9X1^2X2 8X1X2 468X1^5X2^2+44198X1^3X2^3+23634X1X2^4 |
| -6X1^4-22X1^2X2 144X1^4X2^2+522X1^2X2^3-6X2^4 3X1^5X2+14X1^3X2^2+3X1X2^3 -24X1^4X2-88X1^2X2^2 -3X1^5-14X1^3X2-3X1X2^2 6X1^2X2+6X2^2 156X1^6X2^2+15218X1^4X2^3+53120X1^2X2^4-582X2^5 |
| -10X1^3X2-2X1X2^2 240X1^3X2^3+40X1X2^4 5X1^4X2^2+5X1^2X2^3 -40X1^3X2^2-8X1X2^3 -5X1^4X2-5X1^2X2^2 8X1X2^2 260X1^5X2^3+24462X1^3X2^4+4106X1X2^5 |
| -X1^4X2-X1^2X2^2 24X1^4X2^3+23X1^2X2^4-X2^5 1/2X1^5X2^2+X1^3X2^3+1/2X1X2^4 -4X1^4X2^2-4X1^2X2^3 -1/2X1^5X2-X1^3X2^2-1/2X1X2^3 X1^2X2^2+X2^3 26X1^6X2^3+2467X1^4X2^4+2344X1^2X2^5-97X2^6 |
rank = 3
-------------------
gr⁹:
cokernel | -6X1^3X2-2X1X2^2 6X1^2+6X2 42X1 24X1^2X2^3+24X2^4 8/5X1X2^3 -2X1X2^3 -24/5X1^2X2^2-24/5X2^3 -8/5X1X2^2 882X1^4X2^3-47628X1^2X2^4-51912X2^5 -1764X1^3X2^3-1806X1X2^4 |
| -30/7X1^2X2^2+20/7X2^3 50/7X1X2 30X1^2+30X2 200/7X1X2^4 8/7X1^2X2^3+8/7X2^4 -10/7X1^2X2^3-10/7X2^4 -40/7X1X2^3 -8/7X1^2X2^2-8/7X2^3 3480X1^3X2^4-59370X1X2^5 90X1^4X2^3-2100X1^2X2^4+60X2^5 |
| -X1^3X2^2+3X1X2^3 X1^2X2+X2^2 42X1X2 4X1^2X2^4+4X2^5 8/5X1X2^4 -2X1X2^4 -4/5X1^2X2^3-4/5X2^4 -8/5X1X2^3 147X1^4X2^4-5103X1^2X2^5-8652X2^6 -189X1^3X2^4-231X1X2^5 |
| 20/21X1^2X2^3+10/21X2^4 -10/21X1X2^2 5X1^2X2+5X2^2 -40/21X1X2^5 4/21X1^2X2^4+4/21X2^5 -5/21X1^2X2^4-5/21X2^5 8/21X1X2^4 -4/21X1^2X2^3-4/21X2^4 335X1^3X2^5+4525X1X2^6 15X1^4X2^4+175X1^2X2^5+10X2^6 |
rank = 2
-------------------
gr¹⁰:
cokernel | 5X1 0 20X1X2^4 -1/3X1X2^3 -5X1^2X2^5 -5X1^3X2^4+5X1X2^5 -20/3X1X2^5 4/3X1X2^4 0 -4/3X1X2^3 -33600X1^3X2^5+10840X1X2^6 |
| X1^2+X2 0 4X1^2X2^4+4X2^5 -1/15X1^2X2^3-1/15X2^4 -X1^3X2^5-X1X2^6 -X1^4X2^4+X2^6 -4/3X1^2X2^5-4/3X2^6 4/15X1^2X2^4+4/15X2^5 0 -4/15X1^2X2^3-4/15X2^4 -6720X1^4X2^5-4552X1^2X2^6+2168X2^7 |
rank = 1
-------------------
Note that $\mathrm{gr}^i = \mathrm{gr}^i_{\mathrm{tf}}$ for $i\leq 7$, so that we know the class in $K$-theory already: $$\sum_{i=0}^7 [\mathrm{gr}^i_{\mathrm{tf}}] = \sum_{i=0}^7 t^{iw}[i+1]_{t^{w-1}} - \sum_{i=1}^4 t^{(i+1)w+2}[i]_{t^{w-1}}$$ For $i=8,9,10$, in order to compute the $K$-theory class we need to perform more subtle analysis.
The class of $\mathrm{gr}^8$¶
Let us have a closer look at the torsion submodule of $\mathrm{gr}^8$:
In [68]:
MMM8.torsionSubmodule()
Out[68]:
subquotient (| 0 0
| 0 0
| 0 0
| 1974 1554X1
| 15792X1 -624X1^2+1152X2
| 17766X1^2+11844X2 -1998X1^3+2556X1X2
| 5922X1^3+21714X1X2 -666X1^4-858X1^2X2+1584X2^2
| 9870X1^2X2+1974X2^2 -1110X1^3X2+1890X1X2^2
| 987X1^3X2+987X1X2^2 -111X1^4X2+153X1^2X2^2+264X2^3
--------------------------------------------------------------------------------
0 0
0 0
0 1456
-742X1 0
364X1^2-484X2 3536X1^2+2912X2
954X1^3-692X1X2 3978X1^3+3978X1X2
318X1^4+806X1^2X2-360X2^2 1326X1^4+6188X1^2X2+2782X2^2
530X1^3X2-374X1X2^2 2210X1^3X2+2210X1X2^2
53X1^4X2-7X1^2X2^2-60X2^3 221X1^4X2+442X1^2X2^2+221X2^3
--------------------------------------------------------------------------------
52 0 |, |
0 128 | |
0 0 | |
-1484X1X2 742X1^2+742X2 | |
1768X1^2X2-240X2^2 -1004X1^3-748X1X2 | |
3078X1^3X2-214X1X2^2 -1674X1^4-1414X1^2X2-28X2^2 | |
1026X1^4X2+3432X1^2X2^2-330X2^3 -558X1^5-2388X1^3X2-1574X1X2^2 | |
1710X1^3X2^2-98X1X2^3 -930X1^4X2-796X1^2X2^2-26X2^3 | |
171X1^4X2^2+116X1^2X2^3-3X2^4 -93X1^5X2-150X1^3X2^2-57X1X2^3 | |
--------------------------------------------------------------------------------
X1^2+X2 0 0 0 0
8X1X2 X1^2+X2 0 0 0
6X1^2X2+6X2^2 8X1X2 X1^2+X2 0 0
8X1X2^2 6X1^2X2+6X2^2 8X1X2 X1^2+X2 0
X1^2X2^2+X2^3 8X1X2^2 6X1^2X2+6X2^2 8X1X2 X1^2+X2
0 X1^2X2^2+X2^3 8X1X2^2 6X1^2X2+6X2^2 8X1X2
0 0 X1^2X2^2+X2^3 8X1X2^2 6X1^2X2+6X2^2
0 0 0 X1^2X2^2+X2^3 8X1X2^2
0 0 0 0 X1^2X2^2+X2^3
--------------------------------------------------------------------------------
X1^4-2X1^2X2+X2^2 |)
0 |
-4X1^4X2+8X1^2X2^2-4X2^3 |
0 |
6X1^4X2^2-12X1^2X2^3+6X2^4 |
0 |
-4X1^4X2^3+8X1^2X2^4-4X2^5 |
0 |
X1^4X2^4-2X1^2X2^5+X2^6 |
The first $4$ vectors of the monomial basis of $I^{8}/I^{9}$ can be expressed in other basis elements up to torsion. After doing this, we have only columns $2,3$ left, with an appropriate multiple of column $1$ times $X_1$ substracted. Let us check this is torsion-free:
In [69]:
temp = macaulay2('matrix{{47*(-624*X1^2+1152*X2)-37*X1*(15792*X1),141*(364*X1^2-484*X2)+53*X1*(15792*X1)},{47*(-1998*X1^3+2556*X1*X2)-37*X1*(17766*X1^2+11844*X2),141*(954*X1^3-692*X1*X2)+53*X1*(17766*X1^2+11844*X2)},{47*(-666*X1^4-858*X1^2*X2+1584*X2^2)-37*X1*(5922*X1^3+21714*X1*X2),141*(318*X1^4+806*X1^2*X2-360*X2^2)+53*X1*(5922*X1^3+21714*X1*X2)},{47*(-1110*X1^3*X2+1890*X1*X2^2)-37*X1*(9870*X1^2*X2+1974*X2^2),141*(530*X1^3*X2-374*X1*X2^2)+53*X1*(9870*X1^2*X2+1974*X2^2)},{47*(-111*X1^4*X2+153*X1^2*X2^2+264*X2^3)-37*X1*(987*X1^3*X2+987*X1*X2^2),141*(53*X1^4*X2-7*X1^2*X2^2-60*X2^3)+53*X1*(987*X1^3*X2+987*X1*X2^2)}}')
modu = temp.coker()
print(modu)
print("Torsion submodule:")
print(modu.torsionSubmodule().prune())
print("Rank:",modu.rank())
cokernel | -613632X1^2+54144X2 888300X1^2-68244X2 |
| -751248X1^3-318096X1X2 1076112X1^3+530160X1X2 |
| -250416X1^4-843744X1^2X2+74448X2^2 358704X1^4+1264488X1^2X2-50760X2^2 |
| -417360X1^3X2+15792X1X2^2 597840X1^3X2+51888X1X2^2 |
| -41736X1^4X2-29328X1^2X2^2+12408X2^3 59784X1^4X2+51324X1^2X2^2-8460X2^3 |
Torsion submodule:
0
Rank: 3
We obtained the expression of our rank $3$ torsion free sheaf as a quotient of rank $5$ free sheaf by rank $2$ free sheaf. The weight of the latter can be read off by looking at the first non-trivial row, so that $$[\mathrm{gr}^{8}_{\mathrm{tf}}] = t^{12w-4}[5]_{t^{w-1}} - 2t^{10w-2}$$
The class of $\mathrm{gr}^9$¶
Let us have a closer look at the torsion submodule of $\mathrm{gr}^9$:
In [70]:
MMM9.torsionSubmodule()
Out[70]:
subquotient (| 0 0 0 0 0
| 0 0 0 0 0
| 0 0 0 0 0
| 0 0 0 0 210
| 5 0 0 0 0
| 0 21 0 105 0
| 6X2 0 126X1^2+126X2 -42X1 126X1^3+210X1X2
| 0 6X2 150X1X2 -30X1^2 210X1^2X2+150X2^2
| X2^2 0 21X1^2X2+21X2^2 -42X1X2 21X1^3X2+105X1X2^2
| 0 X2^2 -10X1X2^2 -5X1^2X2 -10X2^3
--------------------------------------------------------------------------------
0 0 35 |, |
105 0 0 | |
0 210 0 | |
0 0 0 | |
0 0 0 | |
0 0 0 | |
-378X1^3X2-420X1X2^2 210X1^2X2-42X2^2 672X1^2X2^2+42X2^3 | |
-480X1^2X2^2-450X2^3 -30X1^3X2+270X1X2^2 30X1^3X2^2+780X1X2^3 | |
-63X1^3X2^2-105X1X2^3 -42X2^3 147X1^2X2^3+42X2^4 | |
25X1^2X2^3+30X2^4 -5X1^3X2^2-25X1X2^3 5X1^3X2^3-45X1X2^4 | |
--------------------------------------------------------------------------------
X1^2+X2 0 0 0 0
8X1X2 X1^2+X2 0 0 0
6X1^2X2+6X2^2 8X1X2 X1^2+X2 0 0
8X1X2^2 6X1^2X2+6X2^2 8X1X2 X1^2+X2 0
X1^2X2^2+X2^3 8X1X2^2 6X1^2X2+6X2^2 8X1X2 X1^2+X2
0 X1^2X2^2+X2^3 8X1X2^2 6X1^2X2+6X2^2 8X1X2
0 0 X1^2X2^2+X2^3 8X1X2^2 6X1^2X2+6X2^2
0 0 0 X1^2X2^2+X2^3 8X1X2^2
0 0 0 0 X1^2X2^2+X2^3
0 0 0 0 0
--------------------------------------------------------------------------------
0 X1^4-2X1^2X2+X2^2 0 |)
0 0 X1^4-2X1^2X2+X2^2 |
0 -4X1^4X2+8X1^2X2^2-4X2^3 0 |
0 0 -4X1^4X2+8X1^2X2^2-4X2^3 |
0 6X1^4X2^2-12X1^2X2^3+6X2^4 0 |
X1^2+X2 0 6X1^4X2^2-12X1^2X2^3+6X2^4 |
8X1X2 -4X1^4X2^3+8X1^2X2^4-4X2^5 0 |
6X1^2X2+6X2^2 0 -4X1^4X2^3+8X1^2X2^4-4X2^5 |
8X1X2^2 X1^4X2^4-2X1^2X2^5+X2^6 0 |
X1^2X2^2+X2^3 0 X1^4X2^4-2X1^2X2^5+X2^6 |
The first $6$ vectors of the monomial basis of $I^{9}/I^{10}$ can be expressed in other basis elements up to torsion. After doing this, we have only column $3$, as well as column $4$ minus five times column $2$, left. Let us check that the resulting module is torsion-free:
In [71]:
temp = macaulay2('matrix{{126*X1^2 + 126*X2,-42*X1},{150*X1*X2,-30*X1^2-30*X2},{21*X1^2*X2+21*X2^2,-42*X1*X2},{-10*X1*X2^2,-5*X1^2*X2-5*X2^2}}')
modu = temp.coker()
print(modu)
print("Torsion submodule:")
print(modu.torsionSubmodule().prune())
print("Rank:",modu.rank())
cokernel | 126X1^2+126X2 -42X1 |
| 150X1X2 -30X1^2-30X2 |
| 21X1^2X2+21X2^2 -42X1X2 |
| -10X1X2^2 -5X1^2X2-5X2^2 |
Torsion submodule:
0
Rank: 2
We obtained the expression of our rank $2$ torsion free sheaf as a quotient of rank $4$ free sheaf by rank $2$ free sheaf. The weight of the latter can be read off by looking at the first non-trivial row, so that $$[\mathrm{gr}^{9}_{\mathrm{tf}}] = t^{15w-6}[4]_{t^{w-1}} - t^{13w-4}[2]_{t^{w-1}}$$
The class of $\mathrm{gr}^{10}$¶
Let us have a closer look at the torsion submodule of $\mathrm{gr}^{10}$:
In [72]:
MMM10.torsionSubmodule()
Out[72]:
subquotient (| 0 0 0 0 0 0 0 0
| 0 0 0 0 0 0 0 0
| 0 0 0 0 0 0 15 0
| 0 0 0 0 0 0 0 30
| 0 0 0 0 0 15 0 0
| 1 0 0 0 0 0 0 0
| 0 0 0 15 0 0 0 0
| 0 30 0 0 0 0 0 0
| 0 0 15 0 0 0 0 0
| 0 5X2 0 0 -5X1 5X1X2^2 -5X1X2^3 -5X1^2X2^2-5X2^3
| 0 0 -X2 X2^2 -X1^2-X2 X1^2X2^2 -X1^2X2^3 -X1^3X2^2-X1X2^3
--------------------------------------------------------------------------------
1 0 |, | X1^2+X2 0 0
0 5 | | 8X1X2 X1^2+X2 0
0 0 | | 6X1^2X2+6X2^2 8X1X2 X1^2+X2
0 0 | | 8X1X2^2 6X1^2X2+6X2^2 8X1X2
0 0 | | X1^2X2^2+X2^3 8X1X2^2 6X1^2X2+6X2^2
0 0 | | 0 X1^2X2^2+X2^3 8X1X2^2
0 0 | | 0 0 X1^2X2^2+X2^3
0 0 | | 0 0 0
0 0 | | 0 0 0
-5X1X2^4 5X1^2X2^3+5X2^4 | | 0 0 0
-X1^2X2^4 X1^3X2^3+X1X2^4 | | 0 0 0
--------------------------------------------------------------------------------
0 0 0 0
0 0 0 0
0 0 0 0
X1^2+X2 0 0 0
8X1X2 X1^2+X2 0 0
6X1^2X2+6X2^2 8X1X2 X1^2+X2 0
8X1X2^2 6X1^2X2+6X2^2 8X1X2 X1^2+X2
X1^2X2^2+X2^3 8X1X2^2 6X1^2X2+6X2^2 8X1X2
0 X1^2X2^2+X2^3 8X1X2^2 6X1^2X2+6X2^2
0 0 X1^2X2^2+X2^3 8X1X2^2
0 0 0 X1^2X2^2+X2^3
--------------------------------------------------------------------------------
X1^4-2X1^2X2+X2^2 0 0
0 X1^4-2X1^2X2+X2^2 0
-4X1^4X2+8X1^2X2^2-4X2^3 0 X1^4-2X1^2X2+X2^2
0 -4X1^4X2+8X1^2X2^2-4X2^3 0
6X1^4X2^2-12X1^2X2^3+6X2^4 0 -4X1^4X2+8X1^2X2^2-4X2^3
0 6X1^4X2^2-12X1^2X2^3+6X2^4 0
-4X1^4X2^3+8X1^2X2^4-4X2^5 0 6X1^4X2^2-12X1^2X2^3+6X2^4
0 -4X1^4X2^3+8X1^2X2^4-4X2^5 0
X1^4X2^4-2X1^2X2^5+X2^6 0 -4X1^4X2^3+8X1^2X2^4-4X2^5
0 X1^4X2^4-2X1^2X2^5+X2^6 0
0 0 X1^4X2^4-2X1^2X2^5+X2^6
--------------------------------------------------------------------------------
|)
|
|
|
|
|
|
|
|
|
|
The first $9$ vectors of the monomial basis of $I^{10}/I^{11}$ can be expressed in other basis elements up to torsion. After doing this, we have only column $5$ left. We obtained the expression of our rank $1$ torsion free sheaf as a quotient of rank $2$ free sheaf by rank $1$ free sheaf. The weight of the latter can be read off by looking at the first non-trivial row, so that $$[\mathrm{gr}^{10}_{\mathrm{tf}}] = t^{19w-9}[2]_{t^{w-1}} - t^{18w-8}$$
Final result¶
Summarizing everything, we get $$m(t) = [11]_{t^{2w-1}} + t^w[10]_{t^{2w-1}} + t^{2w}[8]_{t^{2w-1}} + t^{3w}[7]_{t^{2w-1}} + t^{4w}[5]_{t^{2w-1}} + t^{5w}[3]_{t^{2w-1}} - t^{2w+2}[4]_{t^{w}} - t^{4w+1}[3]_{t^{w}} - 2t^{10w-2} - t^{13w-4}[2]_{t^{w-1}} - t^{18w-8}$$ Moreover, recall that in this case $w=2$, so that the multiplicity can be rewritten in a more compact form: $$m(t) = 1 + t^2[16]_t + t^{14}[4]_t + t^{19}[3]_t + t^{20}[2]_t + 2t^{24}+ t^{25}[6]_t -2t^{28}$$
Case $x^3y^4$¶
Setup¶
The usual multiplicity is again $32$. We have $\deg_Y p_1 = 4$, $\deg_Y p_2 = 8$, therefore we may start from $\mathrm{gr}^4$. Furthermore, simply by looking at ranks of modules and comparing with non-equivariant multiplicity, we only need to go until $\mathrm{gr}^{10}$. In order to shorten the computations, let us observe that as in the previous case, nothing interesting will happen until $\mathrm{gr}^{8}$. Namely, $\mathrm{gr}^i = \mathrm{gr}^i_{\mathrm{tf}}$ for $i\leq 7$, so that: $$\sum_{i=0}^7 [\mathrm{gr}^i_{\mathrm{tf}}] = \sum_{i=0}^7 t^{iw}[i+1]_{t^{w-1}} - \sum_{i=1}^4 t^{iw+3}[i]_{t^{w-1}}$$
In [73]:
P1 = 3*Y1^4*X1*X2 + Y1^4*X1^3 + 4*Y1^3*Y2*X2^2 + 12*Y1^3*Y2*X1^2*X2 + 18*Y1^2*Y2^2*X1*X2^2 + 6*Y1^2*Y2^2*X1^3*X2 + 4*Y1*Y2^3*X2^3 + 12*Y1*Y2^3*X1^2*X2^2 + 3*Y2^4*X1*X2^3 + Y2^4*X1^3*X2^2
P2 = -Y1^8*X2^3 + 3*Y1^8*X1^2*X2^2 - 3*Y1^8*X1^4*X2 + Y1^8*X1^6 + 4*Y1^6*Y2^2*X2^4 - 12*Y1^6*Y2^2*X1^2*X2^3 + 12*Y1^6*Y2^2*X1^4*X2^2 - 4*Y1^6*Y2^2*X1^6*X2 - 6*Y1^4*Y2^4*X2^5 + 18*Y1^4*Y2^4*X1^2*X2^4 - 18*Y1^4*Y2^4*X1^4*X2^3 + 6*Y1^4*Y2^4*X1^6*X2^2 + 4*Y1^2*Y2^6*X2^6 - 12*Y1^2*Y2^6*X1^2*X2^5 + 12*Y1^2*Y2^6*X1^4*X2^4 - 4*Y1^2*Y2^6*X1^6*X2^3 - Y2^8*X2^7 + 3*Y2^8*X1^2*X2^6 - 3*Y2^8*X1^4*X2^5 + Y2^8*X1^6*X2^4
#arr4 = [P1]
#arr5 = [Y1*P1,Y2*P1]
#arr6 = [Y1^2*P1,Y1*Y2*P1,Y2^2*P1]
#arr7 = [Y1^3*P1,Y1^2*Y2*P1,Y1*Y2^2*P1,Y2^3*P1]
arr8 = [Y1^4*P1,Y1^3*Y2*P1,Y1^2*Y2^2*P1,Y1*Y2^3*P1,Y2^4*P1,P2]
arr9 = [Y1^5*P1,Y1^4*Y2*P1,Y1^3*Y2^2*P1,Y1^2*Y2^3*P1,Y1*Y2^4*P1,Y2^5*P1,Y1*P2,Y2*P2]
arr10 = [Y1^6*P1,Y1^5*Y2*P1,Y1^4*Y2^2*P1,Y1^3*Y2^3*P1,Y1^2*Y2^4*P1,Y1*Y2^5*P1,Y2^6*P1,Y1^2*P2,Y1*Y2*P2,Y2^2*P2]
Rp = macaulay2('QQ[X1,X2]')
#mat = macaulay2(toMacau(arr4,4))
#MMM4 = mat.coker()
#print("gr⁴:")
#print(MMM4)
#mat = macaulay2(toMacau(arr5,5))
#MMM5 = mat.coker()
#print("gr⁵:")
#print(MMM5)
#mat = macaulay2(toMacau(arr6,6))
#MMM6 = mat.coker()
#print("gr⁶:")
#print(MMM6)
#mat = macaulay2(toMacau(arr7,7))
#MMM7 = mat.coker()
#print("gr⁷:")
#print(MMM7)
mat = macaulay2(toMacau(arr8,8))
MMM8 = mat.coker()
print("gr⁸:")
print(MMM8)
mat = macaulay2(toMacau(arr9,9))
MMM9 = mat.coker()
print("gr⁹:")
print(MMM9)
mat = macaulay2(toMacau(arr10,10))
MMM10 = mat.coker()
print("gr¹⁰:")
print(MMM10)
gr⁸:
cokernel | X1^3+3X1X2 0 0 0 0 X1^6-3X1^4X2+3X1^2X2^2-X2^3 |
| 12X1^2X2+4X2^2 X1^3+3X1X2 0 0 0 0 |
| 6X1^3X2+18X1X2^2 12X1^2X2+4X2^2 X1^3+3X1X2 0 0 -4X1^6X2+12X1^4X2^2-12X1^2X2^3+4X2^4 |
| 12X1^2X2^2+4X2^3 6X1^3X2+18X1X2^2 12X1^2X2+4X2^2 X1^3+3X1X2 0 0 |
| X1^3X2^2+3X1X2^3 12X1^2X2^2+4X2^3 6X1^3X2+18X1X2^2 12X1^2X2+4X2^2 X1^3+3X1X2 6X1^6X2^2-18X1^4X2^3+18X1^2X2^4-6X2^5 |
| 0 X1^3X2^2+3X1X2^3 12X1^2X2^2+4X2^3 6X1^3X2+18X1X2^2 12X1^2X2+4X2^2 0 |
| 0 0 X1^3X2^2+3X1X2^3 12X1^2X2^2+4X2^3 6X1^3X2+18X1X2^2 -4X1^6X2^3+12X1^4X2^4-12X1^2X2^5+4X2^6 |
| 0 0 0 X1^3X2^2+3X1X2^3 12X1^2X2^2+4X2^3 0 |
| 0 0 0 0 X1^3X2^2+3X1X2^3 X1^6X2^4-3X1^4X2^5+3X1^2X2^6-X2^7 |
gr⁹:
cokernel | X1^3+3X1X2 0 0 0 0 0 X1^6-3X1^4X2+3X1^2X2^2-X2^3 0 |
| 12X1^2X2+4X2^2 X1^3+3X1X2 0 0 0 0 0 X1^6-3X1^4X2+3X1^2X2^2-X2^3 |
| 6X1^3X2+18X1X2^2 12X1^2X2+4X2^2 X1^3+3X1X2 0 0 0 -4X1^6X2+12X1^4X2^2-12X1^2X2^3+4X2^4 0 |
| 12X1^2X2^2+4X2^3 6X1^3X2+18X1X2^2 12X1^2X2+4X2^2 X1^3+3X1X2 0 0 0 -4X1^6X2+12X1^4X2^2-12X1^2X2^3+4X2^4 |
| X1^3X2^2+3X1X2^3 12X1^2X2^2+4X2^3 6X1^3X2+18X1X2^2 12X1^2X2+4X2^2 X1^3+3X1X2 0 6X1^6X2^2-18X1^4X2^3+18X1^2X2^4-6X2^5 0 |
| 0 X1^3X2^2+3X1X2^3 12X1^2X2^2+4X2^3 6X1^3X2+18X1X2^2 12X1^2X2+4X2^2 X1^3+3X1X2 0 6X1^6X2^2-18X1^4X2^3+18X1^2X2^4-6X2^5 |
| 0 0 X1^3X2^2+3X1X2^3 12X1^2X2^2+4X2^3 6X1^3X2+18X1X2^2 12X1^2X2+4X2^2 -4X1^6X2^3+12X1^4X2^4-12X1^2X2^5+4X2^6 0 |
| 0 0 0 X1^3X2^2+3X1X2^3 12X1^2X2^2+4X2^3 6X1^3X2+18X1X2^2 0 -4X1^6X2^3+12X1^4X2^4-12X1^2X2^5+4X2^6 |
| 0 0 0 0 X1^3X2^2+3X1X2^3 12X1^2X2^2+4X2^3 X1^6X2^4-3X1^4X2^5+3X1^2X2^6-X2^7 0 |
| 0 0 0 0 0 X1^3X2^2+3X1X2^3 0 X1^6X2^4-3X1^4X2^5+3X1^2X2^6-X2^7 |
gr¹⁰:
cokernel | X1^3+3X1X2 0 0 0 0 0 0 X1^6-3X1^4X2+3X1^2X2^2-X2^3 0 0 |
| 12X1^2X2+4X2^2 X1^3+3X1X2 0 0 0 0 0 0 X1^6-3X1^4X2+3X1^2X2^2-X2^3 0 |
| 6X1^3X2+18X1X2^2 12X1^2X2+4X2^2 X1^3+3X1X2 0 0 0 0 -4X1^6X2+12X1^4X2^2-12X1^2X2^3+4X2^4 0 X1^6-3X1^4X2+3X1^2X2^2-X2^3 |
| 12X1^2X2^2+4X2^3 6X1^3X2+18X1X2^2 12X1^2X2+4X2^2 X1^3+3X1X2 0 0 0 0 -4X1^6X2+12X1^4X2^2-12X1^2X2^3+4X2^4 0 |
| X1^3X2^2+3X1X2^3 12X1^2X2^2+4X2^3 6X1^3X2+18X1X2^2 12X1^2X2+4X2^2 X1^3+3X1X2 0 0 6X1^6X2^2-18X1^4X2^3+18X1^2X2^4-6X2^5 0 -4X1^6X2+12X1^4X2^2-12X1^2X2^3+4X2^4 |
| 0 X1^3X2^2+3X1X2^3 12X1^2X2^2+4X2^3 6X1^3X2+18X1X2^2 12X1^2X2+4X2^2 X1^3+3X1X2 0 0 6X1^6X2^2-18X1^4X2^3+18X1^2X2^4-6X2^5 0 |
| 0 0 X1^3X2^2+3X1X2^3 12X1^2X2^2+4X2^3 6X1^3X2+18X1X2^2 12X1^2X2+4X2^2 X1^3+3X1X2 -4X1^6X2^3+12X1^4X2^4-12X1^2X2^5+4X2^6 0 6X1^6X2^2-18X1^4X2^3+18X1^2X2^4-6X2^5 |
| 0 0 0 X1^3X2^2+3X1X2^3 12X1^2X2^2+4X2^3 6X1^3X2+18X1X2^2 12X1^2X2+4X2^2 0 -4X1^6X2^3+12X1^4X2^4-12X1^2X2^5+4X2^6 0 |
| 0 0 0 0 X1^3X2^2+3X1X2^3 12X1^2X2^2+4X2^3 6X1^3X2+18X1X2^2 X1^6X2^4-3X1^4X2^5+3X1^2X2^6-X2^7 0 -4X1^6X2^3+12X1^4X2^4-12X1^2X2^5+4X2^6 |
| 0 0 0 0 0 X1^3X2^2+3X1X2^3 12X1^2X2^2+4X2^3 0 X1^6X2^4-3X1^4X2^5+3X1^2X2^6-X2^7 0 |
| 0 0 0 0 0 0 X1^3X2^2+3X1X2^3 0 0 X1^6X2^4-3X1^4X2^5+3X1^2X2^6-X2^7 |
In [74]:
#NN = MMM4 / MMM4.torsionSubmodule()
#print("gr⁴:")
#print(NN.prune())
#print("rank =",NN.rank(),"\n-------------------")
#NN = MMM5 / MMM5.torsionSubmodule()
#print("gr⁵:")
#print(NN.prune())
#print("rank =",NN.rank(),"\n-------------------")
#NN = MMM6 / MMM6.torsionSubmodule()
#print("gr⁶:")
#print(NN.prune())
#print("rank =",NN.rank(),"\n-------------------")
#NN = MMM7 / MMM7.torsionSubmodule()
#print("gr⁷:")
#print(NN.prune())
#print("rank =",NN.rank(),"\n-------------------")
NN = MMM8 / MMM8.torsionSubmodule()
print("gr⁸:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM9 / MMM9.torsionSubmodule()
print("gr⁹:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM10 / MMM10.torsionSubmodule()
print("gr¹⁰:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
gr⁸:
cokernel | 3X1^2+X2 3X1^2X2+X2^2 -72X1^2X2^2-24X2^3 -X1^3X2-3X1X2^2 X1^3+3X1X2 0 8X1 0 |
| X1^3+3X1X2 X1^3X2+3X1X2^2 -25X1^3X2^2-75X1X2^3 -12X1^2X2^2-4X2^3 12X1^2X2+4X2^2 X1^3+3X1X2 36X1^2+12X2 0 |
| 3X1^2X2+X2^2 3X1^2X2^2+X2^3 -84X1^2X2^3-28X2^4 -6X1^3X2^2-18X1X2^3 6X1^3X2+18X1X2^2 12X1^2X2+4X2^2 27X1^3+57X1X2 0 |
| 0 0 -6X1^3X2^3-18X1X2^4 -12X1^2X2^3-4X2^4 12X1^2X2^2+4X2^3 6X1^3X2+18X1X2^2 6X1^4+54X1^2X2+12X2^2 0 |
| 0 0 -12X1^2X2^4-4X2^5 -X1^3X2^3-3X1X2^4 X1^3X2^2+3X1X2^3 12X1^2X2^2+4X2^3 15X1^3X2+13X1X2^2 0 |
| 0 0 -X1^3X2^4-3X1X2^5 0 0 X1^3X2^2+3X1X2^3 X1^4X2+3X1^2X2^2 0 |
rank = 3
-------------------
gr⁹:
cokernel | -6/5X1^4X2-9/5X1^2X2^2+3/5X2^3 6X1^3+18X1X2 63X1^2+21X2 24X1^3X2^3+72X1X2^4 12/5X1^2X2^3+4/5X2^4 -3X1^2X2^3-X2^4 -24/5X1^3X2^2-72/5X1X2^3 -12/5X1^2X2^2-4/5X2^3 882X1^5X2^3-53508X1^3X2^4-177534X1X2^5 7938X1^6X2^3+441882X1^4X2^4+3043278X1^2X2^5+655494X2^6 |
| -9/7X1^3X2^2+13/7X1X2^3 75/7X1^2X2+25/7X2^2 30X1^3+90X1X2 300/7X1^2X2^4+100/7X2^5 8/7X1^3X2^3+24/7X1X2^4 -10/7X1^3X2^3-30/7X1X2^4 -60/7X1^2X2^3-20/7X2^4 -8/7X1^3X2^2-24/7X1X2^3 60X1^6X2^3+3330X1^4X2^4-100575X1^2X2^5-35475X2^6 45855X1^5X2^4+1663935X1^3X2^5+3018510X1X2^6 |
| -1/5X1^4X2^2+6/5X1^2X2^3+3/5X2^4 X1^3X2+3X1X2^2 63X1^2X2+21X2^2 4X1^3X2^4+12X1X2^5 12/5X1^2X2^4+4/5X2^5 -3X1^2X2^4-X2^5 -4/5X1^3X2^3-12/5X1X2^4 -12/5X1^2X2^3-4/5X2^4 252X1^5X2^4-5733X1^3X2^5-28539X1X2^6 1323X1^6X2^4+129087X1^4X2^5+2164428X1^2X2^6+655494X2^7 |
| 2/7X1^3X2^3+10/21X1X2^4 -5/7X1^2X2^2-5/21X2^3 5X1^3X2+15X1X2^2 -20/7X1^2X2^5-20/21X2^6 4/21X1^3X2^4+4/7X1X2^5 -5/21X1^3X2^4-5/7X1X2^5 4/7X1^2X2^4+4/21X2^5 -4/21X1^3X2^3-4/7X1X2^4 10X1^6X2^4+240X1^4X2^5+7965X1^2X2^6+2365X2^7 4335X1^5X2^5+129745X1^3X2^6+454260X1X2^7 |
rank = 2
-------------------
gr¹⁰:
cokernel | -1/2X1^2X2^2-1/6X2^3 15X1^2+5X2 0 -1/2X1^2X2^3-1/6X2^4 -10X1^2X2^5-10/3X2^6 2X1^2X2^4+2/3X2^5 0 -2X1^2X2^3-2/3X2^4 0 -75X1^5X2^4+65X1^3X2^5+30X1X2^6 -7650X1^4X2^5+9405X1^2X2^6+3985X2^7 -10080X1^5X2^5+4004X1^3X2^6+7364/3X1X2^7 |
| -1/15X1^3X2^2-1/5X1X2^3 2X1^3+6X1X2 0 -1/15X1^3X2^3-1/5X1X2^4 -4/3X1^3X2^5-4X1X2^6 4/15X1^3X2^4+4/5X1X2^5 0 -4/15X1^3X2^3-4/5X1X2^4 0 -10X1^6X2^4-18X1^4X2^5+36X1^2X2^6 -1020X1^5X2^5-1466X1^3X2^6+4782X1X2^7 -1344X1^6X2^5-45752/15X1^4X2^6+14728/5X1^2X2^7 |
rank = 1
-------------------
As before we perform more careful analysis for $i=8,9,10$.
The class of $\mathrm{gr}^8$¶
In [75]:
MMM8.torsionSubmodule()
Out[75]:
subquotient (| 0 0
| 0 0
| 273 0
| 0 -11743056X1
| 546X2 -52843752X1^2-17614584X2
| 0 -39632814X1^3-83669274X1X2
| 273X2^2 -8807292X1^4-79265628X1^2X2-17614584X2^2
| 0 -22018230X1^3X2-19082466X1X2^2
| 0 -1467882X1^4X2-4403646X1^2X2^2
--------------------------------------------------------------------------------
0
1793376
0
-85526831X1^2-392301X2
-223663050X1^3-110720766X1X2
-157575483X1^4-378894828X1^2X2-2858193X2^2
-35016774X1^5-318849804X1^3X2-81130062X1X2^2
-87541935X1^4X2-83267353X1^2X2^2-672516X2^3
-5836129X1^5X2-18124860X1^3X2^2-1849419X1X2^3
--------------------------------------------------------------------------------
0
0
0
-3151305X1^2-485227X2
-8147628X1^3-4095396X1X2
-5722731X1^4-13922118X1^2X2-613599X2^2
-1271718X1^5-11638020X1^3X2-3121110X1X2^2
-3179295X1^4X2-3140505X1^2X2^2-128372X2^3
-211953X1^5X2-667952X1^3X2^2-96279X1X2^3
--------------------------------------------------------------------------------
0
0
0
10135044X1^2+1481068X2
27178173X1^3+13232439X1X2
19209960X1^4+46003464X1^2X2+1816368X2^2
4268880X1^5+38922870X1^3X2+10046610X1X2^2
10672200X1^4X2+10255140X1^2X2^2+335300X2^3
711480X1^5X2+2218265X1^3X2^2+251475X1X2^3
--------------------------------------------------------------------------------
149448 |, | X1^3+3X1X2
0 | | 12X1^2X2+4X2^2
0 | | 6X1^3X2+18X1X2^2
19616253X1^3+6941111X1X2 | | 12X1^2X2^2+4X2^3
50993766X1^4+43109298X1^2X2+8014032X2^2 | | X1^3X2^2+3X1X2^3
35897985X1^5+99726588X1^3X2+31667691X1X2^2 | | 0
7977330X1^6+75575940X1^4X2+45059850X1^2X2^2+5921760X2^3 | | 0
19943325X1^5X2+26324595X1^3X2^2+6961300X1X2^3 | | 0
1329555X1^6X2+4618660X1^4X2^2+1889985X1^2X2^3+149448X2^4 | | 0
--------------------------------------------------------------------------------
0 0 0 0
X1^3+3X1X2 0 0 0
12X1^2X2+4X2^2 X1^3+3X1X2 0 0
6X1^3X2+18X1X2^2 12X1^2X2+4X2^2 X1^3+3X1X2 0
12X1^2X2^2+4X2^3 6X1^3X2+18X1X2^2 12X1^2X2+4X2^2 X1^3+3X1X2
X1^3X2^2+3X1X2^3 12X1^2X2^2+4X2^3 6X1^3X2+18X1X2^2 12X1^2X2+4X2^2
0 X1^3X2^2+3X1X2^3 12X1^2X2^2+4X2^3 6X1^3X2+18X1X2^2
0 0 X1^3X2^2+3X1X2^3 12X1^2X2^2+4X2^3
0 0 0 X1^3X2^2+3X1X2^3
--------------------------------------------------------------------------------
X1^6-3X1^4X2+3X1^2X2^2-X2^3 |)
0 |
-4X1^6X2+12X1^4X2^2-12X1^2X2^3+4X2^4 |
0 |
6X1^6X2^2-18X1^4X2^3+18X1^2X2^4-6X2^5 |
0 |
-4X1^6X2^3+12X1^4X2^4-12X1^2X2^5+4X2^6 |
0 |
X1^6X2^4-3X1^4X2^5+3X1^2X2^6-X2^7 |
The first $3$ vectors of the monomial basis of $I^{8}/I^{9}$ can be expressed in other basis elements up to torsion. After doing this, we have only columns $2,4,5$ left. Let us check this is torsion-free:
In [76]:
temp = macaulay2('matrix{{-11743056*X1,-3151305*X1^2-485227*X2,10135044*X1^2+1481068*X2},{-52843752*X1^2-17614584*X2,-8147628*X1^3-4095396*X1*X2,27178173*X1^3+13232439*X1*X2},{-39632814*X1^3-83669274*X1*X2,-5722731*X1^4-13922118*X1^2*X2-613599*X2^2,19209960*X1^4+46003464*X1^2*X2+1816368*X2^2},{-8807292*X1^4-79265628*X1^2*X2-17614584*X2^2,-1271718*X1^5-11638020*X1^3*X2-3121110*X1*X2^2,4268880*X1^5+38922870*X1^3*X2+10046610*X1*X2^2},{-22018230*X1^3*X2-19082466*X1*X2^2,-3179295*X1^4*X2-3140505*X1^2*X2^2-128372*X2^3,10672200*X1^4*X2+10255140*X1^2*X2^2+335300*X2^3},{-1467882*X1^4*X2-4403646*X1^2*X2^2,-211953*X1^5*X2-667952*X1^3*X2^2-96279*X1*X2^3,711480*X1^5*X2+2218265*X1^3*X2^2+251475*X1*X2^3}}')
modu = temp.coker()
print(modu)
print("Torsion submodule:")
print(modu.torsionSubmodule().prune())
print("Rank:",modu.rank())
cokernel | -11743056X1 -3151305X1^2-485227X2 10135044X1^2+1481068X2 |
| -52843752X1^2-17614584X2 -8147628X1^3-4095396X1X2 27178173X1^3+13232439X1X2 |
| -39632814X1^3-83669274X1X2 -5722731X1^4-13922118X1^2X2-613599X2^2 19209960X1^4+46003464X1^2X2+1816368X2^2 |
| -8807292X1^4-79265628X1^2X2-17614584X2^2 -1271718X1^5-11638020X1^3X2-3121110X1X2^2 4268880X1^5+38922870X1^3X2+10046610X1X2^2 |
| -22018230X1^3X2-19082466X1X2^2 -3179295X1^4X2-3140505X1^2X2^2-128372X2^3 10672200X1^4X2+10255140X1^2X2^2+335300X2^3 |
| -1467882X1^4X2-4403646X1^2X2^2 -211953X1^5X2-667952X1^3X2^2-96279X1X2^3 711480X1^5X2+2218265X1^3X2^2+251475X1X2^3 |
Torsion submodule:
0
Rank: 3
We obtained the expression of our rank $3$ torsion free sheaf as a quotient of rank $6$ free sheaf by rank $3$ free sheaf. The weight of the latter can be read off by looking at the first non-trivial row, so that $$[\mathrm{gr}^{8}_{\mathrm{tf}}] = t^{11w-3}[6]_{t^{w-1}} - t^{10w-2} - 2t^{9w-1}$$
The class of $\mathrm{gr}^9$¶
In [77]:
MMM9.torsionSubmodule()
Out[77]:
subquotient (| 0 0 0 0
| 0 0 0 0
| 0 0 0 0
| 0 0 0 0
| 5 0 0 0
| 0 21 0 -840X1
| 6X2 0 -126X1^3-378X1X2 189X1^2+63X2
| 0 6X2 -225X1^2X2-75X2^2 90X1^3+30X1X2
| X2^2 0 -21X1^3X2-63X1X2^2 189X1^2X2+63X2^2
| 0 X2^2 15X1^2X2^2+5X2^3 15X1^3X2+5X1X2^2
--------------------------------------------------------------------------------
0 0
0 280
0 0
280 0
0 0
840X1^2+280X2 -840X1^2X2-280X2^2
-525X1^3-1071X1X2 756X1^5+4473X1^3X2+6111X1X2^2
-90X1^4-630X1^2X2 1440X1^4X2+4080X1^2X2^2
-245X1^3X2-231X1X2^2 126X1^5X2+903X1^3X2^2+1071X1X2^3
-15X1^4X2+35X1^2X2^2 -75X1^4X2^2-265X1^2X2^3
--------------------------------------------------------------------------------
0 280 |, |
0 0 | |
280 0 | |
0 0 | |
0 0 | |
-280X1^3-840X1X2 1680X1^3X2+5040X1X2^2 | |
-315X1^4-945X1^2X2-336X2^2 -6048X1^4X2-18144X1^2X2^2-5040X2^3 | |
30X1^5-585X1^3X2-225X1X2^2 -180X1^5X2-10665X1^3X2^2-3375X1X2^3 | |
-56X2^3 -1323X1^4X2^2-3969X1^2X2^3-840X2^4 | |
5X1^5X2+60X1^3X2^2+15X1X2^3 -30X1^5X2^2+585X1^3X2^3+225X1X2^4 | |
--------------------------------------------------------------------------------
X1^3+3X1X2 0 0 0
12X1^2X2+4X2^2 X1^3+3X1X2 0 0
6X1^3X2+18X1X2^2 12X1^2X2+4X2^2 X1^3+3X1X2 0
12X1^2X2^2+4X2^3 6X1^3X2+18X1X2^2 12X1^2X2+4X2^2 X1^3+3X1X2
X1^3X2^2+3X1X2^3 12X1^2X2^2+4X2^3 6X1^3X2+18X1X2^2 12X1^2X2+4X2^2
0 X1^3X2^2+3X1X2^3 12X1^2X2^2+4X2^3 6X1^3X2+18X1X2^2
0 0 X1^3X2^2+3X1X2^3 12X1^2X2^2+4X2^3
0 0 0 X1^3X2^2+3X1X2^3
0 0 0 0
0 0 0 0
--------------------------------------------------------------------------------
0 0 X1^6-3X1^4X2+3X1^2X2^2-X2^3
0 0 0
0 0 -4X1^6X2+12X1^4X2^2-12X1^2X2^3+4X2^4
0 0 0
X1^3+3X1X2 0 6X1^6X2^2-18X1^4X2^3+18X1^2X2^4-6X2^5
12X1^2X2+4X2^2 X1^3+3X1X2 0
6X1^3X2+18X1X2^2 12X1^2X2+4X2^2 -4X1^6X2^3+12X1^4X2^4-12X1^2X2^5+4X2^6
12X1^2X2^2+4X2^3 6X1^3X2+18X1X2^2 0
X1^3X2^2+3X1X2^3 12X1^2X2^2+4X2^3 X1^6X2^4-3X1^4X2^5+3X1^2X2^6-X2^7
0 X1^3X2^2+3X1X2^3 0
--------------------------------------------------------------------------------
0 |)
X1^6-3X1^4X2+3X1^2X2^2-X2^3 |
0 |
-4X1^6X2+12X1^4X2^2-12X1^2X2^3+4X2^4 |
0 |
6X1^6X2^2-18X1^4X2^3+18X1^2X2^4-6X2^5 |
0 |
-4X1^6X2^3+12X1^4X2^4-12X1^2X2^5+4X2^6 |
0 |
X1^6X2^4-3X1^4X2^5+3X1^2X2^6-X2^7 |
The first $6$ vectors of the monomial basis of $I^{9}/I^{10}$ can be expressed in other basis elements up to torsion. After doing this, we have only column $3$, as well as column $4$ plus $40$ times column $2$, left. Let us check that the resulting module is torsion-free:
In [78]:
temp = macaulay2('matrix{{-126*X1^3-378*X1*X2,189*X1^2+63*X2},{-225*X1^2*X2-75*X2^2,90*X1^3+30*X1*X2+240*X2},{-21*X1^3*X2-63*X1*X2^2,189*X1^2*X2+63*X2^2},{15*X1^2*X2^2+5*X2^3,15*X1^3*X2+5*X1*X2^2+40*X2^2}}')
modu = temp.coker()
print(modu)
print("Torsion submodule:")
print(modu.torsionSubmodule().prune())
print("Rank:",modu.rank())
cokernel | -126X1^3-378X1X2 189X1^2+63X2 |
| -225X1^2X2-75X2^2 90X1^3+30X1X2+240X2 |
| -21X1^3X2-63X1X2^2 189X1^2X2+63X2^2 |
| 15X1^2X2^2+5X2^3 15X1^3X2+5X1X2^2+40X2^2 |
Torsion submodule:
0
Rank: 2
We obtained the expression of our rank $2$ torsion free sheaf as a quotient of rank $4$ free sheaf by rank $2$ free sheaf. The weight of the latter can be read off by looking at the first non-trivial row, so that $$[\mathrm{gr}^{9}_{\mathrm{tf}}] = t^{15w-6}[4]_{t^{w-1}} - t^{12w-3}[2]_{t^{w-1}}$$
The class of $\mathrm{gr}^{10}$¶
In [79]:
MMM10.torsionSubmodule()
Out[79]:
subquotient (| 0 0 0 0 0 0
| 0 0 0 0 0 0
| 0 0 0 0 0 0
| 0 0 0 0 0 0
| 0 0 0 0 0 240
| 1 0 0 0 0 0
| 0 0 15 0 0 0
| 0 0 0 90 0 0
| 0 15 0 0 0 0
| 0 0 0 -45X1^2 45X1^2+15X2 -135X1^3X2-45X1X2^2
| 0 -X2 X2^2 -6X1^3-18X1X2 6X1^3+18X1X2 -18X1^4X2-54X1^2X2^2-16X2^3
--------------------------------------------------------------------------------
0 0
0 0
240 0
0 240
0 0
0 0
0 0
0 0
0 0
135X1^3X2^2+45X1X2^3 45X1^4X2+135X1^2X2^2
18X1^4X2^2+54X1^2X2^3+16X2^4 6X1^5X2+34X1^3X2^2+48X1X2^3
--------------------------------------------------------------------------------
16 0 |, |
0 40 | |
0 0 | |
0 0 | |
0 0 | |
0 0 | |
0 0 | |
0 0 | |
0 0 | |
135X1^3X2^3+45X1X2^4 -45X1^4X2^2-135X1^2X2^3 | |
18X1^4X2^3+54X1^2X2^4+16X2^5 -6X1^5X2^2-34X1^3X2^3-48X1X2^4 | |
--------------------------------------------------------------------------------
X1^3+3X1X2 0 0 0
12X1^2X2+4X2^2 X1^3+3X1X2 0 0
6X1^3X2+18X1X2^2 12X1^2X2+4X2^2 X1^3+3X1X2 0
12X1^2X2^2+4X2^3 6X1^3X2+18X1X2^2 12X1^2X2+4X2^2 X1^3+3X1X2
X1^3X2^2+3X1X2^3 12X1^2X2^2+4X2^3 6X1^3X2+18X1X2^2 12X1^2X2+4X2^2
0 X1^3X2^2+3X1X2^3 12X1^2X2^2+4X2^3 6X1^3X2+18X1X2^2
0 0 X1^3X2^2+3X1X2^3 12X1^2X2^2+4X2^3
0 0 0 X1^3X2^2+3X1X2^3
0 0 0 0
0 0 0 0
0 0 0 0
--------------------------------------------------------------------------------
0 0 0
0 0 0
0 0 0
0 0 0
X1^3+3X1X2 0 0
12X1^2X2+4X2^2 X1^3+3X1X2 0
6X1^3X2+18X1X2^2 12X1^2X2+4X2^2 X1^3+3X1X2
12X1^2X2^2+4X2^3 6X1^3X2+18X1X2^2 12X1^2X2+4X2^2
X1^3X2^2+3X1X2^3 12X1^2X2^2+4X2^3 6X1^3X2+18X1X2^2
0 X1^3X2^2+3X1X2^3 12X1^2X2^2+4X2^3
0 0 X1^3X2^2+3X1X2^3
--------------------------------------------------------------------------------
X1^6-3X1^4X2+3X1^2X2^2-X2^3 0
0 X1^6-3X1^4X2+3X1^2X2^2-X2^3
-4X1^6X2+12X1^4X2^2-12X1^2X2^3+4X2^4 0
0 -4X1^6X2+12X1^4X2^2-12X1^2X2^3+4X2^4
6X1^6X2^2-18X1^4X2^3+18X1^2X2^4-6X2^5 0
0 6X1^6X2^2-18X1^4X2^3+18X1^2X2^4-6X2^5
-4X1^6X2^3+12X1^4X2^4-12X1^2X2^5+4X2^6 0
0 -4X1^6X2^3+12X1^4X2^4-12X1^2X2^5+4X2^6
X1^6X2^4-3X1^4X2^5+3X1^2X2^6-X2^7 0
0 X1^6X2^4-3X1^4X2^5+3X1^2X2^6-X2^7
0 0
--------------------------------------------------------------------------------
0 |)
0 |
X1^6-3X1^4X2+3X1^2X2^2-X2^3 |
0 |
-4X1^6X2+12X1^4X2^2-12X1^2X2^3+4X2^4 |
0 |
6X1^6X2^2-18X1^4X2^3+18X1^2X2^4-6X2^5 |
0 |
-4X1^6X2^3+12X1^4X2^4-12X1^2X2^5+4X2^6 |
0 |
X1^6X2^4-3X1^4X2^5+3X1^2X2^6-X2^7 |
The first $9$ vectors of the monomial basis of $I^{10}/I^{11}$ can be expressed in other basis elements up to torsion. After doing this, we have only column $5$ left. We obtained the expression of our rank $1$ torsion free sheaf as a quotient of rank $2$ free sheaf by rank $1$ free sheaf. The weight of the latter can be read off by looking at the first non-trivial row, so that $$[\mathrm{gr}^{10}_{\mathrm{tf}}] = t^{19w-9}[2]_{t^{w-1}} - t^{17w-7}$$
Final result¶
Summarizing everything, we get $$ \begin{align*} m(t) & = [11]_{t^{2w-1}} + t^w[10]_{t^{2w-1}} + t^{2w}[8]_{t^{2w-1}} + t^{3w}[7]_{t^{2w-1}} + t^{4w}[5]_{t^{2w-1}} + t^{5w}[4]_{t^{2w-1}} + t^{6w}[2]_{t^{2w-1}}\\ & - t^{w+3}[4]_{t^{w}} - t^{3w+2}[3]_{t^{w}} - t^{5w+1}[2]_{t^{w}} - 2t^{9w-1} - t^{10w-2} - t^{12w-3}[2]_{t^{w-1}} - t^{17w-7} \end{align*} $$ Moreover, recall that in this case $w=3$, so that the multiplicity can be rewritten in a more compact form: $$m(t) = [7]_{t^5} + t^3[5]_{t^5} + t^{21} + t^{18}[4]_{t^2} + t^{23}[5]_{t^2} + t^{30}[6]_{t^2} + t^{39}[4]_{t^2} + t^{48}[2]_{t^2} - t^{26} - t^{44}$$
Case $x^4y^5$¶
Setup¶
The usual multiplicity is $50$, the biggest one. We have $\deg_Y p_1 = 5$, $\deg_Y p_2 = 10$, therefore we may start from $\mathrm{gr}^5$. Furthermore, simply by looking at ranks of modules and comparing with non-equivariant multiplicity, we only need to go until $\mathrm{gr}^{13}$. In order to shorten the computations, let us observe that as in the previous case, nothing interesting will happen until $\mathrm{gr}^{10}$. Namely, $\mathrm{gr}^i = \mathrm{gr}^i_{\mathrm{tf}}$ for $i\leq 9$, so that: $$\sum_{i=0}^9 [\mathrm{gr}^i_{\mathrm{tf}}] = \sum_{i=0}^9 t^{iw}[i+1]_{t^{w-1}} - \sum_{i=1}^5 t^{iw+4}[i]_{t^{w-1}}$$
In [80]:
P1 = Y1^5*X2^2 + 6*Y1^5*X1^2*X2 + Y1^5*X1^4 + 20*Y1^4*Y2*X1*X2^2 + 20*Y1^4*Y2*X1^3*X2 + 10*Y1^3*Y2^2*X2^3 + 60*Y1^3*Y2^2*X1^2*X2^2 + 10*Y1^3*Y2^2*X1^4*X2 + 40*Y1^2*Y2^3*X1*X2^3 + 40*Y1^2*Y2^3*X1^3*X2^2 + 5*Y1*Y2^4*X2^4 + 30*Y1*Y2^4*X1^2*X2^3 + 5*Y1*Y2^4*X1^4*X2^2 + 4*Y2^5*X1*X2^4 + 4*Y2^5*X1^3*X2^3
P2 = Y1^10*X2^4 - 4*Y1^10*X1^2*X2^3 + 6*Y1^10*X1^4*X2^2 - 4*Y1^10*X1^6*X2 + Y1^10*X1^8 - 5*Y1^8*Y2^2*X2^5 + 20*Y1^8*Y2^2*X1^2*X2^4 - 30*Y1^8*Y2^2*X1^4*X2^3 + 20*Y1^8*Y2^2*X1^6*X2^2 - 5*Y1^8*Y2^2*X1^8*X2 + 10*Y1^6*Y2^4*X2^6 - 40*Y1^6*Y2^4*X1^2*X2^5 + 60*Y1^6*Y2^4*X1^4*X2^4 - 40*Y1^6*Y2^4*X1^6*X2^3 + 10*Y1^6*Y2^4*X1^8*X2^2 - 10*Y1^4*Y2^6*X2^7 + 40*Y1^4*Y2^6*X1^2*X2^6 - 60*Y1^4*Y2^6*X1^4*X2^5 + 40*Y1^4*Y2^6*X1^6*X2^4 - 10*Y1^4*Y2^6*X1^8*X2^3 + 5*Y1^2*Y2^8*X2^8 - 20*Y1^2*Y2^8*X1^2*X2^7 + 30*Y1^2*Y2^8*X1^4*X2^6 - 20*Y1^2*Y2^8*X1^6*X2^5 + 5*Y1^2*Y2^8*X1^8*X2^4 - Y2^10*X2^9 + 4*Y2^10*X1^2*X2^8 - 6*Y2^10*X1^4*X2^7 + 4*Y2^10*X1^6*X2^6 - Y2^10*X1^8*X2^5
#arr5 = [P1]
#arr6 = [Y1*P1,Y2*P1]
#arr7 = [Y1^2*P1,Y1*Y2*P1,Y2^2*P1]
#arr8 = [Y1^3*P1,Y1^2*Y2*P1,Y1*Y2^2*P1,Y2^3*P1]
arr9 = [Y1^4*P1,Y1^3*Y2*P1,Y1^2*Y2^2*P1,Y1*Y2^3*P1,Y2^4*P1]
arr10 = [Y1^5*P1,Y1^4*Y2*P1,Y1^3*Y2^2*P1,Y1^2*Y2^3*P1,Y1*Y2^4*P1,Y2^5*P1,P2]
arr11 = [Y1^6*P1,Y1^5*Y2*P1,Y1^4*Y2^2*P1,Y1^3*Y2^3*P1,Y1^2*Y2^4*P1,Y1*Y2^5*P1,Y2^6*P1,Y1*P2,Y2*P2]
arr12 = [Y1^7*P1,Y1^6*Y2*P1,Y1^5*Y2^2*P1,Y1^4*Y2^3*P1,Y1^3*Y2^4*P1,Y1^2*Y2^5*P1,Y1*Y2^6*P1,Y2^7*P1,Y1^2*P2,Y1*Y2*P2,Y2^2*P2]
arr13 = [Y1^8*P1,Y1^7*Y2*P1,Y1^6*Y2^2*P1,Y1^5*Y2^3*P1,Y1^4*Y2^4*P1,Y1^3*Y2^5*P1,Y1^2*Y2^6*P1,Y1*Y2^7*P1,Y2^8*P1,Y1^3*P2,Y1^2*Y2*P2,Y1*Y2^2*P2,Y2^3*P2]
Rp = macaulay2('QQ[X1,X2]')
#mat = macaulay2(toMacau(arr5,5))
#MMM5 = mat.coker()
#print("gr⁵:")
#print(MMM5)
#mat = macaulay2(toMacau(arr6,6))
#MMM6 = mat.coker()
#print("gr⁶:")
#print(MMM6)
#mat = macaulay2(toMacau(arr7,7))
#MMM7 = mat.coker()
#print("gr⁷:")
#print(MMM7)
#mat = macaulay2(toMacau(arr8,8))
#MMM8 = mat.coker()
#print("gr⁸:")
#print(MMM8)
mat = macaulay2(toMacau(arr9,9))
MMM9 = mat.coker()
print("gr⁹:")
print(MMM9)
mat = macaulay2(toMacau(arr10,10))
MMM10 = mat.coker()
print("gr¹⁰:")
print(MMM10)
mat = macaulay2(toMacau(arr11,11))
MMM11 = mat.coker()
print("gr¹¹:")
print(MMM11)
mat = macaulay2(toMacau(arr12,12))
MMM12 = mat.coker()
print("gr¹²:")
print(MMM12)
mat = macaulay2(toMacau(arr13,13))
MMM13 = mat.coker()
print("gr¹³:")
print(MMM13)
gr⁹:
cokernel | X1^4+6X1^2X2+X2^2 0 0 0 0 |
| 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 0 0 |
| 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 0 |
| 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 |
| 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 |
| 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 |
| 0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 |
| 0 0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 |
| 0 0 0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 |
| 0 0 0 0 4X1^3X2^3+4X1X2^4 |
gr¹⁰:
cokernel | X1^4+6X1^2X2+X2^2 0 0 0 0 0 X1^8-4X1^6X2+6X1^4X2^2-4X1^2X2^3+X2^4 |
| 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 0 0 0 0 |
| 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 0 0 -5X1^8X2+20X1^6X2^2-30X1^4X2^3+20X1^2X2^4-5X2^5 |
| 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 0 0 |
| 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 10X1^8X2^2-40X1^6X2^3+60X1^4X2^4-40X1^2X2^5+10X2^6 |
| 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 |
| 0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 -10X1^8X2^3+40X1^6X2^4-60X1^4X2^5+40X1^2X2^6-10X2^7 |
| 0 0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 0 |
| 0 0 0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 5X1^8X2^4-20X1^6X2^5+30X1^4X2^6-20X1^2X2^7+5X2^8 |
| 0 0 0 0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 0 |
| 0 0 0 0 0 4X1^3X2^3+4X1X2^4 -X1^8X2^5+4X1^6X2^6-6X1^4X2^7+4X1^2X2^8-X2^9 |
gr¹¹:
cokernel | X1^4+6X1^2X2+X2^2 0 0 0 0 0 0 X1^8-4X1^6X2+6X1^4X2^2-4X1^2X2^3+X2^4 0 |
| 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 0 0 0 0 0 X1^8-4X1^6X2+6X1^4X2^2-4X1^2X2^3+X2^4 |
| 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 0 0 0 -5X1^8X2+20X1^6X2^2-30X1^4X2^3+20X1^2X2^4-5X2^5 0 |
| 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 0 0 0 -5X1^8X2+20X1^6X2^2-30X1^4X2^3+20X1^2X2^4-5X2^5 |
| 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 0 10X1^8X2^2-40X1^6X2^3+60X1^4X2^4-40X1^2X2^5+10X2^6 0 |
| 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 0 10X1^8X2^2-40X1^6X2^3+60X1^4X2^4-40X1^2X2^5+10X2^6 |
| 0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 -10X1^8X2^3+40X1^6X2^4-60X1^4X2^5+40X1^2X2^6-10X2^7 0 |
| 0 0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 0 -10X1^8X2^3+40X1^6X2^4-60X1^4X2^5+40X1^2X2^6-10X2^7 |
| 0 0 0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 5X1^8X2^4-20X1^6X2^5+30X1^4X2^6-20X1^2X2^7+5X2^8 0 |
| 0 0 0 0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 0 5X1^8X2^4-20X1^6X2^5+30X1^4X2^6-20X1^2X2^7+5X2^8 |
| 0 0 0 0 0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 -X1^8X2^5+4X1^6X2^6-6X1^4X2^7+4X1^2X2^8-X2^9 0 |
| 0 0 0 0 0 0 4X1^3X2^3+4X1X2^4 0 -X1^8X2^5+4X1^6X2^6-6X1^4X2^7+4X1^2X2^8-X2^9 |
gr¹²:
cokernel | X1^4+6X1^2X2+X2^2 0 0 0 0 0 0 0 X1^8-4X1^6X2+6X1^4X2^2-4X1^2X2^3+X2^4 0 0 |
| 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 0 0 0 0 0 0 X1^8-4X1^6X2+6X1^4X2^2-4X1^2X2^3+X2^4 0 |
| 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 0 0 0 0 -5X1^8X2+20X1^6X2^2-30X1^4X2^3+20X1^2X2^4-5X2^5 0 X1^8-4X1^6X2+6X1^4X2^2-4X1^2X2^3+X2^4 |
| 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 0 0 0 0 -5X1^8X2+20X1^6X2^2-30X1^4X2^3+20X1^2X2^4-5X2^5 0 |
| 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 0 0 10X1^8X2^2-40X1^6X2^3+60X1^4X2^4-40X1^2X2^5+10X2^6 0 -5X1^8X2+20X1^6X2^2-30X1^4X2^3+20X1^2X2^4-5X2^5 |
| 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 0 0 10X1^8X2^2-40X1^6X2^3+60X1^4X2^4-40X1^2X2^5+10X2^6 0 |
| 0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 -10X1^8X2^3+40X1^6X2^4-60X1^4X2^5+40X1^2X2^6-10X2^7 0 10X1^8X2^2-40X1^6X2^3+60X1^4X2^4-40X1^2X2^5+10X2^6 |
| 0 0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 -10X1^8X2^3+40X1^6X2^4-60X1^4X2^5+40X1^2X2^6-10X2^7 0 |
| 0 0 0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 5X1^8X2^4-20X1^6X2^5+30X1^4X2^6-20X1^2X2^7+5X2^8 0 -10X1^8X2^3+40X1^6X2^4-60X1^4X2^5+40X1^2X2^6-10X2^7 |
| 0 0 0 0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 0 5X1^8X2^4-20X1^6X2^5+30X1^4X2^6-20X1^2X2^7+5X2^8 0 |
| 0 0 0 0 0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 -X1^8X2^5+4X1^6X2^6-6X1^4X2^7+4X1^2X2^8-X2^9 0 5X1^8X2^4-20X1^6X2^5+30X1^4X2^6-20X1^2X2^7+5X2^8 |
| 0 0 0 0 0 0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 0 -X1^8X2^5+4X1^6X2^6-6X1^4X2^7+4X1^2X2^8-X2^9 0 |
| 0 0 0 0 0 0 0 4X1^3X2^3+4X1X2^4 0 0 -X1^8X2^5+4X1^6X2^6-6X1^4X2^7+4X1^2X2^8-X2^9 |
gr¹³:
cokernel | X1^4+6X1^2X2+X2^2 0 0 0 0 0 0 0 0 X1^8-4X1^6X2+6X1^4X2^2-4X1^2X2^3+X2^4 0 0 0 |
| 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 0 0 0 0 0 0 0 X1^8-4X1^6X2+6X1^4X2^2-4X1^2X2^3+X2^4 0 0 |
| 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 0 0 0 0 0 -5X1^8X2+20X1^6X2^2-30X1^4X2^3+20X1^2X2^4-5X2^5 0 X1^8-4X1^6X2+6X1^4X2^2-4X1^2X2^3+X2^4 0 |
| 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 0 0 0 0 0 -5X1^8X2+20X1^6X2^2-30X1^4X2^3+20X1^2X2^4-5X2^5 0 X1^8-4X1^6X2+6X1^4X2^2-4X1^2X2^3+X2^4 |
| 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 0 0 0 10X1^8X2^2-40X1^6X2^3+60X1^4X2^4-40X1^2X2^5+10X2^6 0 -5X1^8X2+20X1^6X2^2-30X1^4X2^3+20X1^2X2^4-5X2^5 0 |
| 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 0 0 0 10X1^8X2^2-40X1^6X2^3+60X1^4X2^4-40X1^2X2^5+10X2^6 0 -5X1^8X2+20X1^6X2^2-30X1^4X2^3+20X1^2X2^4-5X2^5 |
| 0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 0 -10X1^8X2^3+40X1^6X2^4-60X1^4X2^5+40X1^2X2^6-10X2^7 0 10X1^8X2^2-40X1^6X2^3+60X1^4X2^4-40X1^2X2^5+10X2^6 0 |
| 0 0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 0 -10X1^8X2^3+40X1^6X2^4-60X1^4X2^5+40X1^2X2^6-10X2^7 0 10X1^8X2^2-40X1^6X2^3+60X1^4X2^4-40X1^2X2^5+10X2^6 |
| 0 0 0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 5X1^8X2^4-20X1^6X2^5+30X1^4X2^6-20X1^2X2^7+5X2^8 0 -10X1^8X2^3+40X1^6X2^4-60X1^4X2^5+40X1^2X2^6-10X2^7 0 |
| 0 0 0 0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 0 5X1^8X2^4-20X1^6X2^5+30X1^4X2^6-20X1^2X2^7+5X2^8 0 -10X1^8X2^3+40X1^6X2^4-60X1^4X2^5+40X1^2X2^6-10X2^7 |
| 0 0 0 0 0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 -X1^8X2^5+4X1^6X2^6-6X1^4X2^7+4X1^2X2^8-X2^9 0 5X1^8X2^4-20X1^6X2^5+30X1^4X2^6-20X1^2X2^7+5X2^8 0 |
| 0 0 0 0 0 0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 0 -X1^8X2^5+4X1^6X2^6-6X1^4X2^7+4X1^2X2^8-X2^9 0 5X1^8X2^4-20X1^6X2^5+30X1^4X2^6-20X1^2X2^7+5X2^8 |
| 0 0 0 0 0 0 0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 0 0 -X1^8X2^5+4X1^6X2^6-6X1^4X2^7+4X1^2X2^8-X2^9 0 |
| 0 0 0 0 0 0 0 0 4X1^3X2^3+4X1X2^4 0 0 0 -X1^8X2^5+4X1^6X2^6-6X1^4X2^7+4X1^2X2^8-X2^9 |
In [81]:
#NN = MMM5 / MMM5.torsionSubmodule()
#print("gr⁵:")
#print(NN.prune())
#print("rank =",NN.rank(),"\n-------------------")
#NN = MMM6 / MMM6.torsionSubmodule()
#print("gr⁶:")
#print(NN.prune())
#print("rank =",NN.rank(),"\n-------------------")
#NN = MMM7 / MMM7.torsionSubmodule()
#print("gr⁷:")
#print(NN.prune())
#print("rank =",NN.rank(),"\n-------------------")
#NN = MMM8 / MMM8.torsionSubmodule()
#print("gr⁸:")
#print(NN.prune())
#print("rank =",NN.rank(),"\n-------------------")
NN = MMM9 / MMM9.torsionSubmodule()
print("gr⁹:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM10 / MMM10.torsionSubmodule()
print("gr¹⁰:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM11 / MMM11.torsionSubmodule()
print("gr¹¹:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM12 / MMM12.torsionSubmodule()
print("gr¹²:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
NN = MMM13 / MMM13.torsionSubmodule()
print("gr¹³:")
print(NN.prune())
print("rank =",NN.rank(),"\n-------------------")
gr⁹:
cokernel | X1^4+6X1^2X2+X2^2 0 0 0 0 |
| 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 0 0 |
| 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 0 |
| 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0 |
| 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 |
| 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 |
| 0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 |
| 0 0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 |
| 0 0 0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 |
| 0 0 0 0 4X1^3X2^3+4X1X2^4 |
rank = 5
-------------------
gr¹⁰:
cokernel | -200X1^3X2^2-200X1X2^3 -2X1^4X2-12X1^2X2^2-2X2^3 20X1^3X2+20X1X2^2 25X1^4+150X1^2X2+25X2^2 0 0 -2500X1^3-2500X1X2 2500X1^2+500X2 |
| -65X1^4X2^2-390X1^2X2^3-65X2^4 -40X1^3X2^2-40X1X2^3 6X1^4X2+36X1^2X2^2+6X2^3 500X1^3X2+500X1X2^2 5X1^4+30X1^2X2+5X2^2 0 -770X1^4-4620X1^2X2-770X2^2 10000X1^3+10000X1X2 |
| -500X1^3X2^3-500X1X2^4 -101/5X1^4X2^2-606/5X1^2X2^3-101/5X2^4 40X1^3X2^2+40X1X2^3 250X1^4X2+1500X1^2X2^2+250X2^3 100X1^3X2+100X1X2^2 X1^4+6X1^2X2+X2^2 -5400X1^3X2-5400X1X2^2 6880X1^4+31280X1^2X2+4880X2^2 |
| -56X1^4X2^3-336X1^2X2^4-56X2^5 -84X1^3X2^3-84X1X2^4 3/5X1^4X2^2+18/5X1^2X2^3+3/5X2^4 1000X1^3X2^2+1000X1X2^3 50X1^4X2+300X1^2X2^2+50X2^3 20X1^3X2+20X1X2^2 -275X1^4X2-1650X1^2X2^2-275X2^3 1485X1^5+26510X1^3X2+19085X1X2^2 |
| -240X1^3X2^4-240X1X2^5 -12X1^4X2^3-72X1^2X2^4-12X2^5 4X1^3X2^3+4X1X2^4 125X1^4X2^2+750X1^2X2^3+125X2^4 200X1^3X2^2+200X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 -1300X1^3X2^2-1300X1X2^3 7200X1^4X2+13700X1^2X2^2+1300X2^3 |
| -17X1^4X2^4-102X1^2X2^5-17X2^6 -16X1^3X2^4-16X1X2^5 -4/5X1^4X2^3-24/5X1^2X2^4-4/5X2^5 100X1^3X2^3+100X1X2^4 25X1^4X2^2+150X1^2X2^3+25X2^4 40X1^3X2^2+40X1X2^3 0 770X1^5X2+1820X1^3X2^2-2030X1X2^3 |
| -20X1^3X2^5-20X1X2^6 -X1^4X2^4-6X1^2X2^5-X2^6 0 0 20X1^3X2^3+20X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 -80X1^3X2^3-80X1X2^4 -3000X1^2X2^3-600X2^4 |
| 2/5X1^4X2^5+12/5X1^2X2^6+2/5X2^7 -4/5X1^3X2^5-4/5X1X2^6 -1/25X1^4X2^4-6/25X1^2X2^5-1/25X2^6 0 0 4X1^3X2^3+4X1X2^4 5X1^4X2^3+30X1^2X2^4+5X2^5 X1^5X2^2-474X1^3X2^3-479X1X2^4 |
rank = 4
-------------------
gr¹¹:
cokernel | 29X1^4+174X1^2X2+29X2^2 0 113X1^4X2^3+678X1^2X2^4+113X2^5 20X1^3X2^3+20X1X2^4 -15X1^4X2^2-90X1^2X2^3-15X2^4 -12X1^3X2^2-12X1X2^3 37/5X1^4X2+222/5X1^2X2^2+37/5X2^3 60X1^3X2+60X1X2^2 3300X1^3+3300X1X2 534187500X1^8X2^3-20440266000X1^6X2^4+197305107912X1^4X2^5+2045735999472X1^2X2^6+345805744812X2^7 -311718750X1^9X2^2+97966575000X1^7X2^3-744230097900X1^5X2^4-8083691634600X1^3X2^5-1318958881350X1X2^6 |
| 80X1^3X2+80X1X2^2 3300X1^3+3300X1X2 340X1^3X2^4+340X1X2^5 9X1^4X2^3+54X1^2X2^4+9X2^5 -44X1^3X2^3-44X1X2^4 -27/5X1^4X2^2-162/5X1^2X2^3-27/5X2^4 20X1^3X2^2+20X1X2^3 27X1^4X2+162X1^2X2^2+27X2^3 1485X1^4+8910X1^2X2+1485X2^2 1794870000X1^7X2^4-71877239280X1^5X2^5+954246498480X1^3X2^6+1037734148880X1X2^7 -1816500000X1^8X2^3+311284176000X1^6X2^4-3746617632240X1^4X2^5-3975434705940X1^2X2^6+26754121260X2^7 |
| 15X1^4X2+90X1^2X2^2+15X2^3 1815X1^4+10890X1^2X2+1815X2^2 74X1^4X2^4+444X1^2X2^5+74X2^6 40X1^3X2^4+40X1X2^5 -46/5X1^4X2^3-276/5X1^2X2^4-46/5X2^5 -24X1^3X2^3-24X1X2^4 18/5X1^4X2^2+108/5X1^2X2^3+18/5X2^4 120X1^3X2^2+120X1X2^3 6600X1^3X2+6600X1X2^2 347221875X1^8X2^4-12793865700X1^6X2^5+122950988226X1^4X2^6+1331368075356X1^2X2^7+226191904851X2^8 -193593750X1^9X2^3+61380033750X1^7X2^4-456674361075X1^5X2^5-5179621067100X1^3X2^6-783768699225X1X2^7 |
| -20/3X1^3X2^2-20/3X1X2^3 7700X1^3X2+7700X1X2^2 40X1^3X2^5+40X1X2^6 14/3X1^4X2^4+28X1^2X2^5+14/3X2^6 -8/3X1^3X2^4-8/3X1X2^5 -14/5X1^4X2^3-84/5X1^2X2^4-14/5X2^5 -8/3X1^3X2^3-8/3X1X2^4 14X1^4X2^2+84X1^2X2^3+14X2^4 770X1^4X2+4620X1^2X2^2+770X2^3 277777500X1^7X2^5-8779538460X1^5X2^6+105965537700X1^3X2^7+120112393500X1X2^8 -531125000X1^8X2^4+38480932000X1^6X2^5-403434145180X1^4X2^6-398743593580X1^2X2^7+13872507320X2^8 |
| 0 957X1^4X2+5742X1^2X2^2+957X2^3 41/5X1^4X2^5+246/5X1^2X2^6+41/5X2^7 4X1^3X2^5+4X1X2^6 -19/25X1^4X2^4-114/25X1^2X2^5-19/25X2^6 -12/5X1^3X2^4-12/5X1X2^5 -3/25X1^4X2^3-18/25X1^2X2^4-3/25X2^5 12X1^3X2^3+12X1X2^4 660X1^3X2^2+660X1X2^3 37393125X1^8X2^5-1419878460X1^6X2^6+68468195334/5X1^4X2^7+734950682004/5X1^2X2^8+124770824409/5X2^9 -17062500X1^9X2^4+6806588250X1^7X2^5-51494435685X1^5X2^6-575856012300X1^3X2^7-88134988125X1X2^8 |
| 28/33X1^3X2^3+28/33X1X2^4 760X1^3X2^2+760X1X2^3 108/11X1^3X2^6+108/11X1X2^7 1/165X1^4X2^5+2/55X1^2X2^6+1/165X2^7 -172/165X1^3X2^5-172/165X1X2^6 -1/275X1^4X2^4-6/275X1^2X2^5-1/275X2^6 4/33X1^3X2^4+4/33X1X2^5 1/55X1^4X2^3+6/55X1^2X2^4+1/55X2^5 X1^4X2^2+6X1^2X2^3+X2^4 45454500X1^7X2^6-2033387748X1^5X2^7+27847953084X1^3X2^8+29933405124X1X2^9 -23275000X1^8X2^5+96408877600/11X1^6X2^6-1220100162124/11X1^4X2^7-1316140741744/11X1^2X2^8+198178676/11X2^9 |
rank = 3
-------------------
gr¹²:
cokernel | 39X1^4X2^2+234X1^2X2^3+39X2^4 -780X1^3X2^2-780X1X2^3 715X1^4+4290X1^2X2+715X2^2 -214500X1^3-214500X1X2 -560/3X1^3X2^5-560/3X1X2^6 -4/3X1^4X2^4-8X1^2X2^5-4/3X2^6 80/3X1^3X2^4+80/3X1X2^5 4/5X1^4X2^3+24/5X1^2X2^4+4/5X2^5 -16X1^3X2^3-16X1X2^4 -4/3X1^4X2^2-8X1^2X2^3-4/3X2^4 5/3X1^4X2^2+10X1^2X2^3+5/3X2^4 1040/3X1^3X2^2+1040/3X1X2^3 66830400X1^7X2^4-48081852000X1^5X2^5+1348027106400X1^3X2^6+1400287824000X1X2^7 -1114863750X1^8X2^4+1592261425755X1^6X2^5-44445462735531X1^4X2^6-48264491743611X1^2X2^7-417869706111X2^8 -1107098685000X1^7X2^5+35464208284890X1^5X2^6-358741244380860X1^3X2^7-396225180320310X1X2^8 |
| 156X1^3X2^3+156X1X2^4 -507/5X1^4X2^2-3042/5X1^2X2^3-507/5X2^4 2860X1^3X2+2860X1X2^2 -27885X1^4-167310X1^2X2-27885X2^2 -364/15X1^4X2^5-728/5X1^2X2^6-364/15X2^7 -16/3X1^3X2^5-16/3X1X2^6 52/15X1^4X2^4+104/5X1^2X2^5+52/15X2^6 16/5X1^3X2^4+16/5X1X2^5 -52/25X1^4X2^3-312/25X1^2X2^4-52/25X2^5 -16/3X1^3X2^3-16/3X1X2^4 20/3X1^3X2^3+20/3X1X2^4 676/15X1^4X2^2+1352/5X1^2X2^3+676/15X2^4 3992625X1^8X2^4-5985432180X1^6X2^5+141424106382X1^4X2^6+1082876754492X1^2X2^7+182166362937X2^8 199119170250X1^7X2^5-4603873480206X1^5X2^6-37052438524530X1^3X2^7-7624209334170X1X2^8 -140540400000X1^8X2^5+3797586898875X1^6X2^6-22502188131291X1^4X2^7-303659850456171X1^2X2^8-51535551454101X2^9 |
| 1131/55X1^4X2^3+6786/55X1^2X2^4+1131/55X2^5 -780/11X1^3X2^3-780/11X1X2^4 377X1^4X2+2262X1^2X2^2+377X2^3 -19500X1^3X2-19500X1X2^2 -560/33X1^3X2^6-560/33X1X2^7 -116/165X1^4X2^5-232/55X1^2X2^6-116/165X2^7 80/33X1^3X2^5+80/33X1X2^6 116/275X1^4X2^4+696/275X1^2X2^5+116/275X2^6 -16/11X1^3X2^4-16/11X1X2^5 -116/165X1^4X2^3-232/55X1^2X2^4-116/165X2^5 29/33X1^4X2^3+58/11X1^2X2^4+29/33X2^5 1040/33X1^3X2^3+1040/33X1X2^4 240196320/11X1^7X2^5-51802732800/11X1^5X2^6+1319634014400/11X1^3X2^7+1395526747680/11X1X2^8 -587837250X1^8X2^5+156215184489X1^6X2^6-20685139360509/5X1^4X2^7-27336784363629/5X1^2X2^8-1101656497929/5X2^9 -111998943000X1^7X2^6+3244103202942X1^5X2^7-32094958510548X1^3X2^8-35932265022258X1X2^9 |
| 156/11X1^3X2^4+156/11X1X2^5 39/5X1^4X2^3+234/5X1^2X2^4+39/5X2^5 260X1^3X2^2+260X1X2^3 2145X1^4X2+12870X1^2X2^2+2145X2^3 28/15X1^4X2^6+56/5X1^2X2^7+28/15X2^8 -16/33X1^3X2^6-16/33X1X2^7 -4/15X1^4X2^5-8/5X1^2X2^6-4/15X2^7 16/55X1^3X2^5+16/55X1X2^6 4/25X1^4X2^4+24/25X1^2X2^5+4/25X2^6 -16/33X1^3X2^4-16/33X1X2^5 20/33X1^3X2^4+20/33X1X2^5 -52/15X1^4X2^3-104/5X1^2X2^4-52/15X2^5 -307125X1^8X2^5+5331313260/11X1^6X2^6-126724567554/11X1^4X2^7-83964096684X1^2X2^8-14012797149X2^9 -16065299250X1^7X2^6+375523461222X1^5X2^7+2591787451578X1^3X2^8+305949726498X1X2^9 10810800000X1^8X2^6-309589158375X1^6X2^7+1849171868487X1^4X2^8+23494151444247X1^2X2^9+3964273188777X2^10 |
rank = 2
-------------------
gr¹³:
cokernel | -29/55X1^4X2^2-174/55X1^2X2^3-29/55X2^4 29X1^4+174X1^2X2+29X2^2 87/55X1^4X2^3+522/55X1^2X2^4+87/55X2^5 0 224/55X1^4X2^6+1344/55X1^2X2^7+224/55X2^8 0 -32/55X1^4X2^5-192/55X1^2X2^6-32/55X2^7 0 96/275X1^4X2^4+576/275X1^2X2^5+96/275X2^6 0 -928/55X1^4X2^3-5568/55X1^2X2^4-928/55X2^5 0 13X1^8X2^5+2652X1^2X2^8+455X2^9 -7055100X1^8X2^5+6629491440X1^6X2^6+49745660965X1^4X2^7+65002524990X1^2X2^8+9721783825X2^9 -2901015000X1^7X2^6+20747893400X1^5X2^7+226022885400X1^3X2^8+38153983400X1X2^9 208X1^5X2^6+1248X1^3X2^7+208X1X2^8 868608000X1^8X2^6-22682431200X1^6X2^7+65047953984X1^4X2^8+1361368847904X1^2X2^9+231543821184X2^10 |
| -116/143X1^3X2^3-116/143X1X2^4 580/13X1^3X2+580/13X1X2^2 348/143X1^3X2^4+348/143X1X2^5 0 896/143X1^3X2^7+896/143X1X2^8 0 -128/143X1^3X2^6-128/143X1X2^7 0 384/715X1^3X2^5+384/715X1X2^6 0 -3712/143X1^3X2^4-3712/143X1X2^5 0 20X1^7X2^6-100X1^5X2^7+580X1^3X2^8+700X1X2^9 -10854000X1^7X2^6+10253487600X1^5X2^7+25220932100X1^3X2^8+14956590500X1X2^9 -4463100000X1^6X2^7+54235336000X1^4X2^8+58698436000X1^2X2^9 320X1^4X2^7+320X1^2X2^8 1336320000X1^7X2^7-41577648000X1^5X2^8+313307295360X1^3X2^9+356221263360X1X2^10 |
rank = 1
-------------------
As before, we perform more careful analysis for $i=10,11,12,13$.
The class of $\mathrm{gr}^{10}$¶
In [82]:
MMM10.torsionSubmodule()
Out[82]:
subquotient (| 0 0 35
| 434417280000 0 0
| 0 105525 0
| 5213007360000X2 0 0
| 0 422100X2 1050X2^2
| 10947315456000X2^2 0 0
| 0 464310X2^2 420X2^3
| 5213007360000X2^3 0 0
| 0 84420X2^3 315X2^4
| 434417280000X2^4 0 0
| 0 4221X2^4 -28X2^5
--------------------------------------------------------------------------------
0
0
0
3384500X1^3+3384500X1X2
1130000X1^4+6780000X1^2X2+1130000X2^2
9062000X1^3X2+9062000X1X2^2
1248035X1^4X2+7488210X1^2X2^2+1248035X2^3
5262900X1^3X2^2+5262900X1X2^3
437870X1^4X2^2+2627220X1^2X2^3+437870X2^4
458600X1^3X2^3+458600X1X2^4
-6769X1^4X2^3-40614X1^2X2^4-6769X2^5
--------------------------------------------------------------------------------
0
0
0
-323500X1^3-323500X1X2
-106250X1^4-637500X1^2X2-106250X2^2
-831000X1^3X2-831000X1X2^2
-101705X1^4X2-610230X1^2X2^2-101705X2^3
-432700X1^3X2^2-432700X1X2^3
-33060X1^4X2^2-198360X1^2X2^3-33060X2^4
-36800X1^3X2^3-36800X1X2^4
647X1^4X2^3+3882X1^2X2^4+647X2^5
--------------------------------------------------------------------------------
0
0
0
-976107500X1^2-195221500X2
-3904430000X1^3-3904430000X1X2
-2841853660X1^4-13146691960X1^2X2-2060967660X2^2
-579807855X1^5-13462760330X1^3X2-10563721055X1X2^2
-4367247800X1^4X2-14685418300X1^2X2^2-2063634100X2^3
-300641110X1^5X2-6934839060X1^3X2^2-5431633510X1X2^3
-778029100X1^4X2^2-3496845600X1^2X2^3-543763300X2^4
-390443X1^5X2^2-437353298X1^3X2^3-435401083X1X2^4
--------------------------------------------------------------------------------
0 |, | X1^4+6X1^2X2+X2^2
0 | | 20X1^3X2+20X1X2^2
0 | | 10X1^4X2+60X1^2X2^2+10X2^3
23807500X1^2+4761500X2 | | 40X1^3X2^2+40X1X2^3
95230000X1^3+95230000X1X2 | | 5X1^4X2^2+30X1^2X2^3+5X2^4
68756060X1^4+317306360X1^2X2+49710060X2^2 | | 4X1^3X2^3+4X1X2^4
14141655X1^5+317211130X1^3X2+246502855X1X2^2 | | 0
100943800X1^4X2+324734300X1^2X2^2+44758100X2^3 | | 0
7332710X1^5X2+146844660X1^3X2^2+110181110X1X2^3 | | 0
16189100X1^4X2^2+68565600X1^2X2^3+10475300X2^4 | | 0
9523X1^5X2^2+8437378X1^3X2^3+8389763X1X2^4 | | 0
--------------------------------------------------------------------------------
0 0 0
X1^4+6X1^2X2+X2^2 0 0
20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0
10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2
40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2
5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3
4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3
0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4
0 0 4X1^3X2^3+4X1X2^4
0 0 0
0 0 0
--------------------------------------------------------------------------------
0 0
0 0
0 0
0 0
X1^4+6X1^2X2+X2^2 0
20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2
10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2
40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3
5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3
4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4
0 4X1^3X2^3+4X1X2^4
--------------------------------------------------------------------------------
X1^8-4X1^6X2+6X1^4X2^2-4X1^2X2^3+X2^4 |)
0 |
-5X1^8X2+20X1^6X2^2-30X1^4X2^3+20X1^2X2^4-5X2^5 |
0 |
10X1^8X2^2-40X1^6X2^3+60X1^4X2^4-40X1^2X2^5+10X2^6 |
0 |
-10X1^8X2^3+40X1^6X2^4-60X1^4X2^5+40X1^2X2^6-10X2^7 |
0 |
5X1^8X2^4-20X1^6X2^5+30X1^4X2^6-20X1^2X2^7+5X2^8 |
0 |
-X1^8X2^5+4X1^6X2^6-6X1^4X2^7+4X1^2X2^8-X2^9 |
The first $3$ vectors of the monomial basis of $I^{10}/I^{11}$ can be expressed in other basis elements up to torsion. After doing this, we have columns $4,5,6,7$ left. They can be checked to have no torsion:
In [83]:
temp = macaulay2('matrix{{3384500*X1^3+3384500*X1*X2,-323500*X1^3-323500*X1*X2,-976107500*X1^2-195221500*X2,23807500*X1^2+4761500*X2},{1130000*X1^4+6780000*X1^2*X2+1130000*X2^2,-106250*X1^4-637500*X1^2*X2-106250*X2^2,-3904430000*X1^3-3904430000*X1*X2,95230000*X1^3+95230000*X1*X2},{9062000*X1^3*X2+9062000*X1*X2^2,-831000*X1^3*X2-831000*X1*X2^2,-2841853660*X1^4-13146691960*X1^2*X2-2060967660*X2^2,68756060*X1^4+317306360*X1^2*X2+49710060*X2^2},{1248035*X1^4*X2+7488210*X1^2*X2^2+1248035*X2^3,-101705*X1^4*X2-610230*X1^2*X2^2-101705*X2^3,-579807855*X1^5-13462760330*X1^3*X2-10563721055*X1*X2^2,14141655*X1^5+317211130*X1^3*X2+246502855*X1*X2^2},{5262900*X1^3*X2^2+5262900*X1*X2^3,-432700*X1^3*X2^2-432700*X1*X2^3,-4367247800*X1^4*X2-14685418300*X1^2*X2^2-2063634100*X2^3,100943800*X1^4*X2+324734300*X1^2*X2^2+44758100*X2^3},{437870*X1^4*X2^2+2627220*X1^2*X2^3+437870*X2^4,-33060*X1^4*X2^2-198360*X1^2*X2^3-33060*X2^4,-300641110*X1^5*X2-6934839060*X1^3*X2^2-5431633510*X1*X2^3,7332710*X1^5*X2+146844660*X1^3*X2^2+110181110*X1*X2^3},{458600*X1^3*X2^3+458600*X1*X2^4,-36800*X1^3*X2^3-36800*X1*X2^4,-778029100*X1^4*X2^2-3496845600*X1^2*X2^3-543763300*X2^4,16189100*X1^4*X2^2+68565600*X1^2*X2^3+10475300*X2^4},{-6769*X1^4*X2^3-40614*X1^2*X2^4-6769*X2^5,647*X1^4*X2^3+3882*X1^2*X2^4+647*X2^5,-390443*X1^5*X2^2-437353298*X1^3*X2^3-435401083*X1*X2^4,9523*X1^5*X2^2+8437378*X1^3*X2^3+8389763*X1*X2^4}}')
modu = temp.coker()
print(modu)
print("Torsion submodule:")
print(modu.torsionSubmodule().prune())
print("Rank:",modu.rank())
cokernel | 3384500X1^3+3384500X1X2 -323500X1^3-323500X1X2 -976107500X1^2-195221500X2 23807500X1^2+4761500X2 |
| 1130000X1^4+6780000X1^2X2+1130000X2^2 -106250X1^4-637500X1^2X2-106250X2^2 -3904430000X1^3-3904430000X1X2 95230000X1^3+95230000X1X2 |
| 9062000X1^3X2+9062000X1X2^2 -831000X1^3X2-831000X1X2^2 -2841853660X1^4-13146691960X1^2X2-2060967660X2^2 68756060X1^4+317306360X1^2X2+49710060X2^2 |
| 1248035X1^4X2+7488210X1^2X2^2+1248035X2^3 -101705X1^4X2-610230X1^2X2^2-101705X2^3 -579807855X1^5-13462760330X1^3X2-10563721055X1X2^2 14141655X1^5+317211130X1^3X2+246502855X1X2^2 |
| 5262900X1^3X2^2+5262900X1X2^3 -432700X1^3X2^2-432700X1X2^3 -4367247800X1^4X2-14685418300X1^2X2^2-2063634100X2^3 100943800X1^4X2+324734300X1^2X2^2+44758100X2^3 |
| 437870X1^4X2^2+2627220X1^2X2^3+437870X2^4 -33060X1^4X2^2-198360X1^2X2^3-33060X2^4 -300641110X1^5X2-6934839060X1^3X2^2-5431633510X1X2^3 7332710X1^5X2+146844660X1^3X2^2+110181110X1X2^3 |
| 458600X1^3X2^3+458600X1X2^4 -36800X1^3X2^3-36800X1X2^4 -778029100X1^4X2^2-3496845600X1^2X2^3-543763300X2^4 16189100X1^4X2^2+68565600X1^2X2^3+10475300X2^4 |
| -6769X1^4X2^3-40614X1^2X2^4-6769X2^5 647X1^4X2^3+3882X1^2X2^4+647X2^5 -390443X1^5X2^2-437353298X1^3X2^3-435401083X1X2^4 9523X1^5X2^2+8437378X1^3X2^3+8389763X1X2^4 |
Torsion submodule:
0
Rank: 4
We obtained the expression of our rank $4$ torsion free sheaf as a quotient of rank $8$ free sheaf by rank $4$ free sheaf. The weight of the latter can be read off by looking at the first non-trivial row, so that $$[\mathrm{gr}^{11}_{\mathrm{tf}}] = t^{13w-3}[8]_{t^{w-1}} - 2t^{10w}[2]_{t^{w-1}}$$
The class of $\mathrm{gr}^{11}$¶
In [84]:
MMM11.torsionSubmodule()
Out[84]:
subquotient (| 0 0 0
| 0 0 0
| 0 0 0
| 0 0 0
| 49225 0 0
| 0 3780480 0
| 127985X2 0 0
| 0 3780480X2 -16500X1^3-16500X1X2
| 68915X2^2 0 -9075X1^4-54450X1^2X2-9075X2^2
| 0 1260160X2^2 -38500X1^3X2-38500X1X2^2
| 5907X2^3 0 -4785X1^4X2-28710X1^2X2^2-4785X2^3
| 0 -22912X2^3 -3800X1^3X2^2-3800X1X2^3
--------------------------------------------------------------------------------
0 0
0 0
0 0
0 -75609600
0 0
0 0
-16500X1^3-16500X1X2 68521200X1^3+68521200X1X2
-7425X1^4-44550X1^2X2-7425X2^2 30834540X1^4+185007240X1^2X2+590700X2^2
-33000X1^3X2-33000X1X2^2 137042400X1^3X2+137042400X1X2^2
-3850X1^4X2-23100X1^2X2^2-3850X2^3 15988280X1^4X2+95929680X1^2X2^2+56313400X2^3
-3300X1^3X2^2-3300X1X2^3 13704240X1^3X2^2+13704240X1X2^3
-5X1^4X2^2-30X1^2X2^3-5X2^4 20764X1^4X2^2+124584X1^2X2^3-4836580X2^4
--------------------------------------------------------------------------------
0
0
75609600
0
0
0
-83879400X1^4-577113900X1^2X2-552304500X2^2
6645375X1^5-257721750X1^3X2-290948625X1X2^2
-35494800X1^4X2-360643800X1^2X2^2-427955880X2^3
3445750X1^5X2-16065500X1^3X2^2-33294250X1X2^3
-4431240X1^4X2^2-41354940X1^2X2^3-40652964X2^4
4475X1^5X2^2-8723950X1^3X2^3-8746325X1X2^4
--------------------------------------------------------------------------------
0
0
0
-75609600X1
0
0
131283075X1^4+445092450X1^2X2+62761875X2^2
30834540X1^5+374344740X1^3X2+189928200X1X2^2
178416150X1^4X2+385284900X1^2X2^2+41373750X2^3
15988280X1^5X2+119304680X1^3X2^2+79688400X1X2^3
18402615X1^4X2^2+41894490X1^2X2^3+4698375X2^4
20764X1^5X2^2+5692084X1^3X2^3+730920X1X2^4
--------------------------------------------------------------------------------
0
6426816000
-56555980800X1
406023552000X1^2+67670592000X2
0
0
-631013945925X1^5-1996569839550X1^3X2-18183961125X1X2^2
-170552220300X1^6-1831043847600X1^4X2-1053272701800X1^2X2^2-136094326500X2^3
-924138713850X1^5X2-1965804611100X1^3X2^2-105681806010X1X2^3
-88434484600X1^6X2-649084207200X1^4X2^2-546553651600X1^2X2^3-67136993000X2^4
-94666465905X1^5X2^2-210096028230X1^3X2^3-15083014353X1X2^4
-114849980X1^6X2^2-23076366960X1^4X2^3+3421955320X1^2X2^4+342409100X2^5
--------------------------------------------------------------------------------
0
0
-75609600X1
0
0
0
83879400X1^5+503276400X1^3X2+478467000X1X2^2
-6645375X1^6+224494875X1^4X2+91587375X1^2X2^2-33226875X2^3
35494800X1^5X2+212968800X1^3X2^2+280280880X1X2^3
-3445750X1^6X2-1163250X1^4X2^2-70078250X1^2X2^3-17228750X2^4
4431240X1^5X2^2+26587440X1^3X2^3+25885464X1X2^4
-4475X1^6X2^2+8701575X1^4X2^3+8612075X1^2X2^4-22375X2^5
--------------------------------------------------------------------------------
584256000 |,
0 |
38560896000X1^2+6426816000X2 |
-65629132800X1^3-65629132800X1X2 |
0 |
0 |
71175215100X1^6+224342543700X1^4X2+81116696100X1^2X2^2-41924932500X2^3 |
30153521970X1^7+237203228790X1^5X2+384914870670X1^3X2^2+123638903850X1X2^3 |
136762870200X1^6X2+367968407400X1^4X2^2+151136363400X1^2X2^3-26926765800X2^4 |
15635159540X1^7X2+118027546780X1^5X2^2+201285100940X1^3X2^3+70775393700X1X2^4 |
13713537420X1^6X2^2+37334951940X1^4X2^3+18580975380X1^2X2^4-2763676740X2^5 |
20305402X1^7X2^2+520948814X1^5X2^3-527326778X1^3X2^4-1064486190X1X2^5 |
--------------------------------------------------------------------------------
| X1^4+6X1^2X2+X2^2 0
| 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2
| 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2
| 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3
| 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3
| 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4
| 0 4X1^3X2^3+4X1X2^4
| 0 0
| 0 0
| 0 0
| 0 0
| 0 0
--------------------------------------------------------------------------------
0 0 0
0 0 0
X1^4+6X1^2X2+X2^2 0 0
20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0
10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2
40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2
5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3
4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3
0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4
0 0 4X1^3X2^3+4X1X2^4
0 0 0
0 0 0
--------------------------------------------------------------------------------
0 0
0 0
0 0
0 0
0 0
X1^4+6X1^2X2+X2^2 0
20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2
10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2
40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3
5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3
4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4
0 4X1^3X2^3+4X1X2^4
--------------------------------------------------------------------------------
X1^8-4X1^6X2+6X1^4X2^2-4X1^2X2^3+X2^4
0
-5X1^8X2+20X1^6X2^2-30X1^4X2^3+20X1^2X2^4-5X2^5
0
10X1^8X2^2-40X1^6X2^3+60X1^4X2^4-40X1^2X2^5+10X2^6
0
-10X1^8X2^3+40X1^6X2^4-60X1^4X2^5+40X1^2X2^6-10X2^7
0
5X1^8X2^4-20X1^6X2^5+30X1^4X2^6-20X1^2X2^7+5X2^8
0
-X1^8X2^5+4X1^6X2^6-6X1^4X2^7+4X1^2X2^8-X2^9
0
--------------------------------------------------------------------------------
0 |)
X1^8-4X1^6X2+6X1^4X2^2-4X1^2X2^3+X2^4 |
0 |
-5X1^8X2+20X1^6X2^2-30X1^4X2^3+20X1^2X2^4-5X2^5 |
0 |
10X1^8X2^2-40X1^6X2^3+60X1^4X2^4-40X1^2X2^5+10X2^6 |
0 |
-10X1^8X2^3+40X1^6X2^4-60X1^4X2^5+40X1^2X2^6-10X2^7 |
0 |
5X1^8X2^4-20X1^6X2^5+30X1^4X2^6-20X1^2X2^7+5X2^8 |
0 |
-X1^8X2^5+4X1^6X2^6-6X1^4X2^7+4X1^2X2^8-X2^9 |
The first $6$ vectors of the monomial basis of $I^{11}/I^{12}$ can be expressed in other basis elements up to torsion. After doing this, we have only columns $3,4,7,9$ left. However, note that $Col_{9} = 4475X_2 Col_4 - X_1 Col_6$, so we can remove it. Furthermore, replacing $Col_7$ with $Col_7-X_1 Col_5$ we can annihilate its 4th coordinate. The remaining three columns can be checked to have no torsion. We obtained the expression of our rank $3$ torsion free sheaf as a quotient of rank $6$ free sheaf by rank $3$ free sheaf. The weight of the latter can be read off by looking at the first non-trivial row, so that $$[\mathrm{gr}^{11}_{\mathrm{tf}}] = t^{17w-6}[6]_{t^{w-1}} - t^{13w-2}[3]_{t^{w-1}}$$
The class of $\mathrm{gr}^{12}$¶
In [85]:
MMM12.torsionSubmodule()
Out[85]:
subquotient (| 0 0 0
| 0 0 0
| 0 0 0
| 0 0 0
| 0 0 0
| 0 0 0
| 0 0 0
| 21120 0 0
| 0 2145 0
| 14080X2 0 275X1^4+1650X1^2X2+275X2^2
| 0 286X2 1100X1^3X2+1100X1X2^2
| 1664X2^2 0 145X1^4X2+870X1^2X2^2+145X2^3
| 0 65X2^2 100X1^3X2^2+100X1X2^3
--------------------------------------------------------------------------------
0 0
0 0
0 0
0 0
0 0
0 0
2860 0
0 21120X1
0 0
1100X1^3+1100X1X2 -16500X1^3-2420X1X2
143X1^4+858X1^2X2+715X2^2 -2145X1^4-12870X1^2X2-2145X2^2
100X1^3X2+100X1X2^2 -1500X1^3X2+164X1X2^2
-11X1^4X2-66X1^2X2^2-115X2^3 165X1^4X2+990X1^2X2^2+165X2^3
--------------------------------------------------------------------------------
0
0
0
0
0
63360
0
126720X1^2+21120X2
0
-97680X1^4-23100X1^2X2-9020X2^2
-12441X1^5-82566X1^3X2-20361X1X2^2
-9744X1^4X2-4980X1^2X2^2-4756X2^3
957X1^5X2+5022X1^3X2^2+237X1X2^3
--------------------------------------------------------------------------------
0
0
0
0
316800
0
0
-2449920X1^3-2449920X1X2
0
1913175X1^5+1806970X1^3X2-102905X1X2^2
248820X1^6+1688676X1^4X2+1439856X1^2X2^2+13200X2^3
173565X1^5X2+117566X1^3X2^2-54259X1X2^3
-19140X1^6X2-130452X1^4X2^2-111312X1^2X2^3+1200X2^4
--------------------------------------------------------------------------------
0
0
0
1584000
36748800X1
0
0
-281550720X1^4-268350720X1^2X2+2640000X2^2
0
219865800X1^6+196960720X1^4X2-15636280X1^2X2^2+110000X2^3
28594995X1^7+192722541X1^5X2+155893221X1^3X2^2-1555125X1X2^3
19946040X1^6X2+12368816X1^4X2^2-7344584X1^2X2^3+58000X2^4
-2199615X1^7X2-14889057X1^5X2^2-12313017X1^3X2^3+119625X1X2^4
--------------------------------------------------------------------------------
0
0
0
0
-316800X1
0
0
2449920X1^4+2449920X1^2X2
0
-1913175X1^6-1802845X1^4X2+127655X1^2X2^2+4125X2^3
-248820X1^7-1688676X1^5X2-1423356X1^3X2^2+3300X1X2^3
-173565X1^6X2-115391X1^4X2^2+67309X1^2X2^3+2175X2^4
19140X1^7X2+130452X1^5X2^2+112812X1^3X2^3+300X1X2^4
--------------------------------------------------------------------------------
0
0
63360
0
-1900800X1^2-316800X2
0
0
14699520X1^5+18078720X1^3X2+3379200X1X2^2
0
-11479050X1^7-13457070X1^5X2-1733270X1^3X2^2+165550X1X2^3
-1492920X1^8-10475256X1^6X2-10873896X1^4X2^2-1970760X1^2X2^3
-1041390X1^7X2-932346X1^5X2^2+238094X1^3X2^3+87290X1X2^4
114840X1^8X2+809112X1^6X2^2+856392X1^4X2^3+154920X1^2X2^4
--------------------------------------------------------------------------------
0
28800
0
0
-1267200X1^3-1267200X1X2
0
0
9799680X1^6+19071360X1^4X2+6631680X1^2X2^2-528000X2^3
0
-7652700X1^8-14451580X1^6X2-4251060X1^4X2^2+787820X1^2X2^3-22000X2^4
-995280X1^9-7696359X1^7X2-11815353X1^5X2^2-3773649X1^3X2^3+311025X1X2^4
-694260X1^8X2-1118324X1^6X2^2+19332X1^4X2^3+235396X1^2X2^4-11600X2^5
76560X1^9X2+594243X1^7X2^2+924381X1^5X2^3+303573X1^3X2^4-23925X1X2^5
--------------------------------------------------------------------------------
5760 |, |
0 | |
0 | |
0 | |
316800X1^4+1900800X1^2X2+316800X2^2 | |
0 | |
0 | |
-2449920X1^7-17149440X1^5X2-16051200X1^3X2^2-1351680X1X2^3 | |
0 | |
1913175X1^9+13281895X1^7X2+11745415X1^5X2^2+229625X1^3X2^3-81070X1X2^4 | |
248820X1^10+3181596X1^8X2+11692692X1^6X2^2+9470340X1^4X2^3+776424X1^2X2^4 | |
173565X1^9X2+1156781X1^7X2^2+721037X1^5X2^3-339725X1^3X2^4-42746X1X2^5 | |
-19140X1^10X2-245292X1^8X2^2-906084X1^6X2^3-748980X1^4X2^4-63048X1^2X2^5 | |
--------------------------------------------------------------------------------
X1^4+6X1^2X2+X2^2 0 0
20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0
10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2
40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2
5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3
4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3
0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4
0 0 4X1^3X2^3+4X1X2^4
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
--------------------------------------------------------------------------------
0 0 0
0 0 0
0 0 0
X1^4+6X1^2X2+X2^2 0 0
20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0
10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2
40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2
5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3
4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3
0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4
0 0 4X1^3X2^3+4X1X2^4
0 0 0
0 0 0
--------------------------------------------------------------------------------
0 0
0 0
0 0
0 0
0 0
0 0
X1^4+6X1^2X2+X2^2 0
20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2
10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2
40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3
5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3
4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4
0 4X1^3X2^3+4X1X2^4
--------------------------------------------------------------------------------
X1^8-4X1^6X2+6X1^4X2^2-4X1^2X2^3+X2^4
0
-5X1^8X2+20X1^6X2^2-30X1^4X2^3+20X1^2X2^4-5X2^5
0
10X1^8X2^2-40X1^6X2^3+60X1^4X2^4-40X1^2X2^5+10X2^6
0
-10X1^8X2^3+40X1^6X2^4-60X1^4X2^5+40X1^2X2^6-10X2^7
0
5X1^8X2^4-20X1^6X2^5+30X1^4X2^6-20X1^2X2^7+5X2^8
0
-X1^8X2^5+4X1^6X2^6-6X1^4X2^7+4X1^2X2^8-X2^9
0
0
--------------------------------------------------------------------------------
0
X1^8-4X1^6X2+6X1^4X2^2-4X1^2X2^3+X2^4
0
-5X1^8X2+20X1^6X2^2-30X1^4X2^3+20X1^2X2^4-5X2^5
0
10X1^8X2^2-40X1^6X2^3+60X1^4X2^4-40X1^2X2^5+10X2^6
0
-10X1^8X2^3+40X1^6X2^4-60X1^4X2^5+40X1^2X2^6-10X2^7
0
5X1^8X2^4-20X1^6X2^5+30X1^4X2^6-20X1^2X2^7+5X2^8
0
-X1^8X2^5+4X1^6X2^6-6X1^4X2^7+4X1^2X2^8-X2^9
0
--------------------------------------------------------------------------------
0 |)
0 |
X1^8-4X1^6X2+6X1^4X2^2-4X1^2X2^3+X2^4 |
0 |
-5X1^8X2+20X1^6X2^2-30X1^4X2^3+20X1^2X2^4-5X2^5 |
0 |
10X1^8X2^2-40X1^6X2^3+60X1^4X2^4-40X1^2X2^5+10X2^6 |
0 |
-10X1^8X2^3+40X1^6X2^4-60X1^4X2^5+40X1^2X2^6-10X2^7 |
0 |
5X1^8X2^4-20X1^6X2^5+30X1^4X2^6-20X1^2X2^7+5X2^8 |
0 |
-X1^8X2^5+4X1^6X2^6-6X1^4X2^7+4X1^2X2^8-X2^9 |
The first $9$ vectors of the monomial basis of $I^{12}/I^{13}$ can be expressed in other basis elements up to torsion. After doing this, we have only columns $3,5,9$ left. However, note that $Col_{9} = 15X_2 Col_3 - X_1 Col_7$, so we can remove it. Furthermore, replacing $Col_5$ with $Col_5-X_1 Col_1$ we can annihilate its 8th coordinate. Let us check the remaining two columns have no torsion:
In [86]:
temp = macaulay2('matrix{{55*X1^4+330*X1^2*X2+55*X2^2,1100*X1^3+1100*X1*X2},{220*X1^3*X2+220*X1*X2^2,143*X1^4+858*X1^2*X2+143*X2^2},{29*X1^4*X2+174*X1^2*X2^2+29*X2^3,100*X1^3*X2+100*X1*X2^2},{20*X1^3*X2^2+20*X1*X2^3,-11*X1^4*X2-66*X1^2*X2^2-11*X2^3}}')
modu = temp.coker()
print(modu)
print("Torsion submodule:")
print(modu.torsionSubmodule().prune())
print("Rank:",modu.rank())
cokernel | 55X1^4+330X1^2X2+55X2^2 1100X1^3+1100X1X2 |
| 220X1^3X2+220X1X2^2 143X1^4+858X1^2X2+143X2^2 |
| 29X1^4X2+174X1^2X2^2+29X2^3 100X1^3X2+100X1X2^2 |
| 20X1^3X2^2+20X1X2^3 -11X1^4X2-66X1^2X2^2-11X2^3 |
Torsion submodule:
0
Rank: 2
We obtained the expression of our rank $2$ torsion free sheaf as a quotient of rank $4$ free sheaf by rank $1$ free sheaf. The weight of the latter can be read off by looking at the first non-trivial row, so that $$[\mathrm{gr}^{12}_{\mathrm{tf}}] = t^{21w-9}[4]_{t^{w-1}} - t^{17w-5}[2]_{t^{w-1}}$$
The class of $\mathrm{gr}^{13}$¶
In [87]:
MMM13.torsionSubmodule()
Out[87]:
subquotient (| 0 0 0 0 0
| 0 0 0 0 0
| 0 0 0 0 0
| 0 0 0 0 0
| 0 0 0 0 0
| 0 0 0 0 0
| 0 0 0 0 0
| 0 0 0 0 0
| 0 0 0 2860 0
| 0 0 715 0 0
| 11 0 0 0 0
| 0 260 0 0 0
| X2 0 0 13X1^4+78X1^2X2+65X2^2 65X1^4+390X1^2X2+65X2^2
| 0 -20X2 29X2^2 20X1^3X2+20X1X2^2 100X1^3X2+100X1X2^2
--------------------------------------------------------------------------------
0 0
0 0
0 0
0 0
0 0
0 0
2860 0
0 -11440
0 0
0 0
0 0
0 0
-39X1^4X2-234X1^2X2^2-195X2^3 -39X1^5-234X1^3X2-39X1X2^2
-60X1^3X2^2-60X1X2^3 -60X1^4X2-60X1^2X2^2+240X2^3
--------------------------------------------------------------------------------
0 0
0 0
0 0
0 0
14300 0
0 11440
0 0
0 0
0 0
0 0
0 0
0 0
377X1^4X2^2+2262X1^2X2^3+1885X2^4 -13X1^5X2-78X1^3X2^2-13X1X2^3
580X1^3X2^3+580X1X2^4 -20X1^4X2^2-20X1^2X2^3+80X2^4
--------------------------------------------------------------------------------
0 0
0 0
0 260
0 0
0 0
0 0
0 0
11440X1 0
0 0
0 0
0 0
0 0
39X1^6+429X1^4X2+1209X1^2X2^2+195X2^3 65X1^6X2^2+442X1^4X2^3+377X1^2X2^4
60X1^5X2+360X1^3X2^2+60X1X2^3 100X1^5X2^3+180X1^3X2^4+80X1X2^5
--------------------------------------------------------------------------------
0 20
0 0
0 0
2288 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
65X1^7X2+767X1^5X2^2+2327X1^3X2^3+377X1X2^4 65X1^6X2^3+442X1^4X2^4+377X1^2X2^5
100X1^6X2^2+680X1^4X2^3+580X1^2X2^4+80X2^5 100X1^5X2^4+180X1^3X2^5+80X1X2^6
--------------------------------------------------------------------------------
0 |, | X1^4+6X1^2X2+X2^2
208 | | 20X1^3X2+20X1X2^2
0 | | 10X1^4X2+60X1^2X2^2+10X2^3
0 | | 40X1^3X2^2+40X1X2^3
0 | | 5X1^4X2^2+30X1^2X2^3+5X2^4
0 | | 4X1^3X2^3+4X1X2^4
0 | | 0
0 | | 0
0 | | 0
0 | | 0
0 | | 0
0 | | 0
-65X1^7X2^2-767X1^5X2^3-2327X1^3X2^4-377X1X2^5 | | 0
-100X1^6X2^3-680X1^4X2^4-580X1^2X2^5-80X2^6 | | 0
--------------------------------------------------------------------------------
0 0 0
X1^4+6X1^2X2+X2^2 0 0
20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0
10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2
40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2
5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3
4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3
0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4
0 0 4X1^3X2^3+4X1X2^4
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
--------------------------------------------------------------------------------
0 0 0
0 0 0
0 0 0
0 0 0
X1^4+6X1^2X2+X2^2 0 0
20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2 0
10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2
40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2
5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3
4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3
0 4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4
0 0 4X1^3X2^3+4X1X2^4
0 0 0
0 0 0
--------------------------------------------------------------------------------
0 0
0 0
0 0
0 0
0 0
0 0
0 0
X1^4+6X1^2X2+X2^2 0
20X1^3X2+20X1X2^2 X1^4+6X1^2X2+X2^2
10X1^4X2+60X1^2X2^2+10X2^3 20X1^3X2+20X1X2^2
40X1^3X2^2+40X1X2^3 10X1^4X2+60X1^2X2^2+10X2^3
5X1^4X2^2+30X1^2X2^3+5X2^4 40X1^3X2^2+40X1X2^3
4X1^3X2^3+4X1X2^4 5X1^4X2^2+30X1^2X2^3+5X2^4
0 4X1^3X2^3+4X1X2^4
--------------------------------------------------------------------------------
X1^8-4X1^6X2+6X1^4X2^2-4X1^2X2^3+X2^4
0
-5X1^8X2+20X1^6X2^2-30X1^4X2^3+20X1^2X2^4-5X2^5
0
10X1^8X2^2-40X1^6X2^3+60X1^4X2^4-40X1^2X2^5+10X2^6
0
-10X1^8X2^3+40X1^6X2^4-60X1^4X2^5+40X1^2X2^6-10X2^7
0
5X1^8X2^4-20X1^6X2^5+30X1^4X2^6-20X1^2X2^7+5X2^8
0
-X1^8X2^5+4X1^6X2^6-6X1^4X2^7+4X1^2X2^8-X2^9
0
0
0
--------------------------------------------------------------------------------
0
X1^8-4X1^6X2+6X1^4X2^2-4X1^2X2^3+X2^4
0
-5X1^8X2+20X1^6X2^2-30X1^4X2^3+20X1^2X2^4-5X2^5
0
10X1^8X2^2-40X1^6X2^3+60X1^4X2^4-40X1^2X2^5+10X2^6
0
-10X1^8X2^3+40X1^6X2^4-60X1^4X2^5+40X1^2X2^6-10X2^7
0
5X1^8X2^4-20X1^6X2^5+30X1^4X2^6-20X1^2X2^7+5X2^8
0
-X1^8X2^5+4X1^6X2^6-6X1^4X2^7+4X1^2X2^8-X2^9
0
0
--------------------------------------------------------------------------------
0
0
X1^8-4X1^6X2+6X1^4X2^2-4X1^2X2^3+X2^4
0
-5X1^8X2+20X1^6X2^2-30X1^4X2^3+20X1^2X2^4-5X2^5
0
10X1^8X2^2-40X1^6X2^3+60X1^4X2^4-40X1^2X2^5+10X2^6
0
-10X1^8X2^3+40X1^6X2^4-60X1^4X2^5+40X1^2X2^6-10X2^7
0
5X1^8X2^4-20X1^6X2^5+30X1^4X2^6-20X1^2X2^7+5X2^8
0
-X1^8X2^5+4X1^6X2^6-6X1^4X2^7+4X1^2X2^8-X2^9
0
--------------------------------------------------------------------------------
0 |)
0 |
0 |
X1^8-4X1^6X2+6X1^4X2^2-4X1^2X2^3+X2^4 |
0 |
-5X1^8X2+20X1^6X2^2-30X1^4X2^3+20X1^2X2^4-5X2^5 |
0 |
10X1^8X2^2-40X1^6X2^3+60X1^4X2^4-40X1^2X2^5+10X2^6 |
0 |
-10X1^8X2^3+40X1^6X2^4-60X1^4X2^5+40X1^2X2^6-10X2^7 |
0 |
5X1^8X2^4-20X1^6X2^5+30X1^4X2^6-20X1^2X2^7+5X2^8 |
0 |
-X1^8X2^5+4X1^6X2^6-6X1^4X2^7+4X1^2X2^8-X2^9 |
The first $12$ vectors of the monomial basis of $I^{13}/I^{14}$ can be expressed in other basis elements up to torsion. After doing this, we have only columns $5$ and $10$ left. However, note that $Col_{10} = 3X_2 Col_5 - X_1 Col_7$, so we can remove it. We obtained the expression of our rank $1$ torsion free sheaf as a quotient of rank $2$ free sheaf by rank $1$ free sheaf. The weight of the latter can be read off by looking at the first non-trivial row, so that $$[\mathrm{gr}^{13}_{\mathrm{tf}}] = t^{25w-12}[2]_{t^{w-1}} - t^{21w-8}$$
Final result¶
Summing everything up and simplifying the formulas as much as possible, we get $$ \begin{align*} m(t) & = [14]_{t^{2w-1}}+t^{w}[13]_{t^{2w-1}}+t^{2w}[11]_{t^{2w-1}}+t^{3w}[10]_{t^{2w-1}}+t^{4w}[8]_{t^{2w-1}}+t^{5w}[7]_{t^{2w-1}}+t^{6w}[5]_{t^{2w-1}}+t^{7w}[4]_{t^{2w-1}}+t^{8w}[2]_{t^{2w-1}}\\ & - t^{w+4}[5]_{t^w} - t^{3w+3}[4]_{t^w} - t^{5w+2}[3]_{t^w} - t^{7w+1}[2]_{t^w} - 2t^{10w}[2]_{t^{w-1}} - t^{13w-2}[3]_{t^{w-1}} - t^{17w-5}[2]_{t^{w-1}} - t^{21w-8} \end{align*} $$ Recalling that $w=2$, we can further write $$m(t) = [40]_t + t^{16}[4]_t + t^{18}[2]_t[2]_{t^4}[2]_{t^5} + t^{33}- t - t^{34} - t^{37}$$ This is a positive polynomial.
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