Use Macaulay2 to compute minimal primes of complicated ideal

Question

I tried computing the minimal primes of a fairly complex ideal using the online Macauley2 interface. I start by letting R=QQ[z] and I=ideal(z^6-1) and work over the polynomial ring

S=R/I[x0,x1,x2,y0,y1,y2]

I want $z=e^{i\pi/3}$ to be the sixth root of unity. I have the following ideal of $S$:

J=ideal(((x0+x1)*(x0+x2)*(x1+x2)-8*x0*x1*x2)*(x0*(x1^2-x2^2)+(z^2)*x1*(x2^2-x0^2)+(z^4)*x2*(x0^2-x1^2))*y0-(x1-x2)*(x2-x0)*(x0-x1)*(x0+(z^4)*x1+(z^2)*x2)*(x1*x2+(z^4)*x0*x2+(z^2)*x0*x1)*y1, (x0*(x1^2-x2^2)+(z^4)*x1*(x2^2-x0^2)+(z^2)*x2*(x0^2-x1^2))*(x0+(z^4)*x1+(z^2)*x2)*(x1*x2+(z^4)*x0*x2+(z^2)*x0*x1)*y1-(x0*(x1^2-x2^2)+(z^2)*x1*(x2^2-x0^2)+(z^4)*x2*(x0^2-x1^2))*(x0+(z^2)*x1+(z^4)*x2)*(x1*x2+(z^2)*x0*x2+(z^4)*x0*x1)*y2, (x1-x2)*(x2-x0)*(x0-x1)*(x0+(z^2)*x1+(z^4)*x2)*(x1*x2+(z^2)*x0*x2+(z^4)*x0*x1)*y2-((x0+x1)*(x0+x2)*(x1+x2)-8*x0*x1*x2)*(x0*(x1^2-x2^2)+(z^4)*x1*(x2^2-x0^2)+(z^2)*x2*(x0^2-x1^2))*y0, (x0+x1)*(x0+x2)*(x1+x2)*((y0+y1)*(y0+y2)*(y1+y2)-8*y0*y1*y2)-((x0*x1*x2)*(y0+y1+y2)*(y0*y1+y0*y2+y1*y2)))

I tried using minimalPrimes J, but the program wouldn't output a result, after working for 30+ minutes. Same for primeDecomposition J. This is a complicated ideal, but I feel like it shouldn't be too crazy for a computer to handle.

Can anyone get this simple code to run easily? Is it possibly because I was using the web interface of Macauley2? If not, are there any simplifications I can make, or more efficient ways of computing the primes of this ideal?

Answer 1

time gens gb J; -- used 14.2537 seconds

time minimalPrimes J -- used 16.5552 seconds

toString oo

{ideal(z-1,x1,x0), 
ideal(z+1,x1,x0), 
ideal(x1,x0,z^2-z+1),
ideal(x1,x0,z^2+z+1), 
ideal(z-1,x2,x1), 
ideal(z+1,x2,x1),
ideal(x2,x1,z^2-z+1), 
ideal(x2,x1,z^2+z+1), 
ideal(z-1,y1-y2,y0-y2,x1),
ideal(z-1,x1,x0+x2), 
ideal(z+1,y1-y2,y0-y2,x1), 
ideal(z+1,x1,x0+x2),
ideal(x1,z^2+z+1,x0*y0+x2*y0+(-z-1)*x0*y1+z*x2*y1+z*x0*y2+(-z-1)*x2*y2,z*x0*y0+z*x2*y0+x0*y1+(-z-1)*x2*y1+(-z-1)*x0*y2+x2*y2,y0^2*y1+y0*y1^2+y0^2*y2-6*y0*y1*y2+y1^2*y2+y0*y2^2+y1*y2^2,2*x0^2*y0+4*x0*x2*y0+2*x2^2*y0-x0^2*y1+x0*x2*y1-x2^2*y1-x0^2*y2+x0*x2*y2-x2^2*y2,x2*y0*y1-x0*y1^2+(z+1)*x2*y1^2+x2*y0*y2+2*x0*y1*y2-3*x2*y1*y2-x0*y2^2-z*x2*y2^2,z*x2*y0*y1-z*x0*y1^2-x2*y1^2+z*x2*y0*y2+2*z*x0*y1*y2-3*z*x2*y1*y2-z*x0*y2^2+(z+1)*x2*y2^2,x0^2*y1+(-2*z-1)*x0*x2*y1-x2^2*y1-x0^2*y2+(-2*z-1)*x0*x2*y2+x2^2*y2,z*x0^2*y1+(z+2)*x0*x2*y1-z*x2^2*y1-z*x0^2*y2+(z+2)*x0*x2*y2+z*x2^2*y2,2*x0*x2*y0*y1-x0^2*y1^2+x0*x2*y1^2-x2^2*y1^2+2*x0*x2*y0*y2+2*x0^2*y1*y2-6*x0*x2*y1*y2+2*x2^2*y1*y2-x0^2*y2^2+x0*x2*y2^2-x2^2*y2^2),
ideal(x1,z^2-z+1,x0*y0+x2*y0+(z-1)*x0*y1-z*x2*y1-z*x0*y2+(z-1)*x2*y2,z*x0*y0+z*x2*y0-x0*y1+(-z+1)*x2*y1+(-z+1)*x0*y2-x2*y2,y0^2*y1+y0*y1^2+y0^2*y2-6*y0*y1*y2+y1^2*y2+y0*y2^2+y1*y2^2,2*x0^2*y0+4*x0*x2*y0+2*x2^2*y0-x0^2*y1+x0*x2*y1-x2^2*y1-x0^2*y2+x0*x2*y2-x2^2*y2,x2*y0*y1-x0*y1^2+(-z+1)*x2*y1^2+x2*y0*y2+2*x0*y1*y2-3*x2*y1*y2-x0*y2^2+z*x2*y2^2,z*x2*y0*y1-z*x0*y1^2+x2*y1^2+z*x2*y0*y2+2*z*x0*y1*y2-3*z*x2*y1*y2-z*x0*y2^2+(z-1)*x2*y2^2,x0^2*y1+(2*z-1)*x0*x2*y1-x2^2*y1-x0^2*y2+(2*z-1)*x0*x2*y2+x2^2*y2,z*x0^2*y1+(z-2)*x0*x2*y1-z*x2^2*y1-z*x0^2*y2+(z-2)*x0*x2*y2+z*x2^2*y2,2*x0*x2*y0*y1-x0^2*y1^2+x0*x2*y1^2-x2^2*y1^2+2*x0*x2*y0*y2+2*x0^2*y1*y2-6*x0*x2*y1*y2+2*x2^2*y1*y2-x0^2*y2^2+x0*x2*y2^2-x2^2*y2^2), 
ideal(z-1,x2,x0),
ideal(z+1,x2,x0), 
ideal(x2,x0,z^2-z+1), 
ideal(x2,x0,z^2+z+1),
ideal(z-1,y1-y2,y0-y2,x0), 
ideal(z-1,x1+x2,x0),
ideal(z+1,y1-y2,y0-y2,x0), 
ideal(z+1,x1+x2,x0),
ideal(x0,z^2+z+1,x1*y0+x2*y0+z*x1*y1+(-z-1)*x2*y1+(-z-1)*x1*y2+z*x2*y2,z*x1*y0+z*x2*y0+(-z-1)*x1*y1+x2*y1+x1*y2+(-z-1)*x2*y2,y0^2*y1+y0*y1^2+y0^2*y2-6*y0*y1*y2+y1^2*y2+y0*y2^2+y1*y2^2,2*x1^2*y0+4*x1*x2*y0+2*x2^2*y0-x1^2*y1+x1*x2*y1-x2^2*y1-x1^2*y2+x1*x2*y2-x2^2*y2,x2*y0*y1-x1*y1^2-z*x2*y1^2+x2*y0*y2+2*x1*y1*y2-3*x2*y1*y2-x1*y2^2+(z+1)*x2*y2^2,z*x2*y0*y1-z*x1*y1^2+(z+1)*x2*y1^2+z*x2*y0*y2+2*z*x1*y1*y2-3*z*x2*y1*y2-z*x1*y2^2-x2*y2^2,x1^2*y1+(2*z+1)*x1*x2*y1-x2^2*y1-x1^2*y2+(2*z+1)*x1*x2*y2+x2^2*y2,z*x1^2*y1+(-z-2)*x1*x2*y1-z*x2^2*y1-z*x1^2*y2+(-z-2)*x1*x2*y2+z*x2^2*y2,2*x1*x2*y0*y1-x1^2*y1^2+x1*x2*y1^2-x2^2*y1^2+2*x1*x2*y0*y2+2*x1^2*y1*y2-6*x1*x2*y1*y2+2*x2^2*y1*y2-x1^2*y2^2+x1*x2*y2^2-x2^2*y2^2),
ideal(x0,z^2-z+1,x1*y0+x2*y0-z*x1*y1+(z-1)*x2*y1+(z-1)*x1*y2-z*x2*y2,z*x1*y0+z*x2*y0+(-z+1)*x1*y1-x2*y1-x1*y2+(-z+1)*x2*y2,y0^2*y1+y0*y1^2+y0^2*y2-6*y0*y1*y2+y1^2*y2+y0*y2^2+y1*y2^2,2*x1^2*y0+4*x1*x2*y0+2*x2^2*y0-x1^2*y1+x1*x2*y1-x2^2*y1-x1^2*y2+x1*x2*y2-x2^2*y2,x2*y0*y1-x1*y1^2+z*x2*y1^2+x2*y0*y2+2*x1*y1*y2-3*x2*y1*y2-x1*y2^2+(-z+1)*x2*y2^2,z*x2*y0*y1-z*x1*y1^2+(z-1)*x2*y1^2+z*x2*y0*y2+2*z*x1*y1*y2-3*z*x2*y1*y2-z*x1*y2^2+x2*y2^2,x1^2*y1+(-2*z+1)*x1*x2*y1-x2^2*y1-x1^2*y2+(-2*z+1)*x1*x2*y2+x2^2*y2,z*x1^2*y1+(-z+2)*x1*x2*y1-z*x2^2*y1-z*x1^2*y2+(-z+2)*x1*x2*y2+z*x2^2*y2,2*x1*x2*y0*y1-x1^2*y1^2+x1*x2*y1^2-x2^2*y1^2+2*x1*x2*y0*y2+2*x1^2*y1*y2-6*x1*x2*y1*y2+2*x2^2*y1*y2-x1^2*y2^2+x1*x2*y2^2-x2^2*y2^2),
ideal(z-1,y1-y2,y0-y2,x2), 
ideal(z-1,x2,x0+x1),
ideal(z+1,y1-y2,y0-y2,x2), 
ideal(z+1,x2,x0+x1),
ideal(x2,z^2-z+1,x0*y0+x1*y0-z*x0*y1+(z-1)*x1*y1+(z-1)*x0*y2-z*x1*y2,z*x0*y0+z*x1*y0+(-z+1)*x0*y1-x1*y1-x0*y2+(-z+1)*x1*y2,y0^2*y1+y0*y1^2+y0^2*y2-6*y0*y1*y2+y1^2*y2+y0*y2^2+y1*y2^2,2*x0^2*y0+4*x0*x1*y0+2*x1^2*y0-x0^2*y1+x0*x1*y1-x1^2*y1-x0^2*y2+x0*x1*y2-x1^2*y2,x1*y0*y1-x0*y1^2+z*x1*y1^2+x1*y0*y2+2*x0*y1*y2-3*x1*y1*y2-x0*y2^2+(-z+1)*x1*y2^2,z*x1*y0*y1-z*x0*y1^2+(z-1)*x1*y1^2+z*x1*y0*y2+2*z*x0*y1*y2-3*z*x1*y1*y2-z*x0*y2^2+x1*y2^2,x0^2*y1+(-2*z+1)*x0*x1*y1-x1^2*y1-x0^2*y2+(-2*z+1)*x0*x1*y2+x1^2*y2,z*x0^2*y1+(-z+2)*x0*x1*y1-z*x1^2*y1-z*x0^2*y2+(-z+2)*x0*x1*y2+z*x1^2*y2,2*x0*x1*y0*y1-x0^2*y1^2+x0*x1*y1^2-x1^2*y1^2+2*x0*x1*y0*y2+2*x0^2*y1*y2-6*x0*x1*y1*y2+2*x1^2*y1*y2-x0^2*y2^2+x0*x1*y2^2-x1^2*y2^2),
ideal(x2,z^2+z+1,x0*y0+x1*y0+z*x0*y1+(-z-1)*x1*y1+(-z-1)*x0*y2+z*x1*y2,z*x0*y0+z*x1*y0+(-z-1)*x0*y1+x1*y1+x0*y2+(-z-1)*x1*y2,y0^2*y1+y0*y1^2+y0^2*y2-6*y0*y1*y2+y1^2*y2+y0*y2^2+y1*y2^2,2*x0^2*y0+4*x0*x1*y0+2*x1^2*y0-x0^2*y1+x0*x1*y1-x1^2*y1-x0^2*y2+x0*x1*y2-x1^2*y2,x1*y0*y1-x0*y1^2-z*x1*y1^2+x1*y0*y2+2*x0*y1*y2-3*x1*y1*y2-x0*y2^2+(z+1)*x1*y2^2,z*x1*y0*y1-z*x0*y1^2+(z+1)*x1*y1^2+z*x1*y0*y2+2*z*x0*y1*y2-3*z*x1*y1*y2-z*x0*y2^2-x1*y2^2,x0^2*y1+(2*z+1)*x0*x1*y1-x1^2*y1-x0^2*y2+(2*z+1)*x0*x1*y2+x1^2*y2,z*x0^2*y1+(-z-2)*x0*x1*y1-z*x1^2*y1-z*x0^2*y2+(-z-2)*x0*x1*y2+z*x1^2*y2,2*x0*x1*y0*y1-x0^2*y1^2+x0*x1*y1^2-x1^2*y1^2+2*x0*x1*y0*y2+2*x0^2*y1*y2-6*x0*x1*y1*y2+2*x1^2*y1*y2-x0^2*y2^2+x0*x1*y2^2-x1^2*y2^2),
ideal(z-1,y2,y1,y0),
ideal(z+1,y2,y1,y0),
ideal(y2,y1,y0,z^2-z+1),
ideal(y2,y1,y0,z^2+z+1),
ideal(z-1,y1,y0,x0*x1+x0*x2+x1*x2),
ideal(z-1,y1,y0,x0+x1+x2),
ideal(z+1,y1,y0,x0*x1+x0*x2+x1*x2),
ideal(z+1,y1,y0,x0+x1+x2),
ideal(y1,y0,z^2+z+1,(z+1)*x0-z*x1-x2),
ideal(y1,y0,z^2+z+1,x0*x1+z*x0*x2+(-z-1)*x1*x2,z*x0*x1+(-z-1)*x0*x2+x1*x2,x0^2*x1^2-x0^2*x1*x2-x0*x1^2*x2+x0^2*x2^2-x0*x1*x2^2+x1^2*x2^2),
ideal(y1,y0,z^2-z+1,(z-1)*x0-z*x1+x2),
ideal(y1,y0,z^2-z+1,x0*x1-z*x0*x2+(z-1)*x1*x2,z*x0*x1+(-z+1)*x0*x2-x1*x2,x0^2*x1^2-x0^2*x1*x2-x0*x1^2*x2+x0^2*x2^2-x0*x1*x2^2+x1^2*x2^2),
ideal(z-1,y2,y0,x0*x1+x0*x2+x1*x2),
ideal(z-1,y2,y0,x0+x1+x2),
ideal(z-1,y1+y2,y0,x0*x1+x0*x2+x1*x2),
ideal(z-1,y1+y2,y0,x0+x1+x2),
ideal(z+1,y2,y0,x0*x1+x0*x2+x1*x2),
ideal(z+1,y2,y0,x0+x1+x2),
ideal(z+1,y1+y2,y0,x0*x1+x0*x2+x1*x2),
ideal(z+1,y1+y2,y0,x0+x1+x2),
ideal(y1+y2,y0,x0-x1,z^2+z+1),
ideal(y1+y2,y0,x0-x1,z^2-z+1),
ideal(y2,y0,z^2+z+1,z*x0+(-z-1)*x1+x2),
ideal(y2,y0,z^2+z+1,x0*x1+(-z-1)*x0*x2+z*x1*x2,z*x0*x1+x0*x2+(-z-1)*x1*x2,x0^2*x1^2-x0^2*x1*x2-x0*x1^2*x2+x0^2*x2^2-x0*x1*x2^2+x1^2*x2^2),
ideal(y1+y2,y0,x1-x2,z^2+z+1),
ideal(y1+y2,y0,x0-x2,z^2+z+1),
ideal(y2,y0,z^2-z+1,z*x0+(-z+1)*x1-x2),
ideal(y2,y0,z^2-z+1,x0*x1+(z-1)*x0*x2-z*x1*x2,z*x0*x1-x0*x2+(-z+1)*x1*x2,x0^2*x1^2-x0^2*x1*x2-x0*x1^2*x2+x0^2*x2^2-x0*x1*x2^2+x1^2*x2^2),
ideal(y1+y2,y0,x1-x2,z^2-z+1), 
ideal(y1+y2,y0,x0-x2,z^2-z+1),
ideal(z-1,x0-x1,2*x1^2*y0^2*y1+3*x1*x2*y0^2*y1+2*x2^2*y0^2*y1+2*x1^2*y0*y1^2+3*x1*x2*y0*y1^2+2*x2^2*y0*y1^2+2*x1^2*y0^2*y2+3*x1*x2*y0^2*y2+2*x2^2*y0^2*y2-12*x1^2*y0*y1*y2-27*x1*x2*y0*y1*y2-12*x2^2*y0*y1*y2+2*x1^2*y1^2*y2+3*x1*x2*y1^2*y2+2*x2^2*y1^2*y2+2*x1^2*y0*y2^2+3*x1*x2*y0*y2^2+2*x2^2*y0*y2^2+2*x1^2*y1*y2^2+3*x1*x2*y1*y2^2+2*x2^2*y1*y2^2),
ideal(z-1,x0-x2,2*x1^2*y0^2*y1+3*x1*x2*y0^2*y1+2*x2^2*y0^2*y1+2*x1^2*y0*y1^2+3*x1*x2*y0*y1^2+2*x2^2*y0*y1^2+2*x1^2*y0^2*y2+3*x1*x2*y0^2*y2+2*x2^2*y0^2*y2-12*x1^2*y0*y1*y2-27*x1*x2*y0*y1*y2-12*x2^2*y0*y1*y2+2*x1^2*y1^2*y2+3*x1*x2*y1^2*y2+2*x2^2*y1^2*y2+2*x1^2*y0*y2^2+3*x1*x2*y0*y2^2+2*x2^2*y0*y2^2+2*x1^2*y1*y2^2+3*x1*x2*y1*y2^2+2*x2^2*y1*y2^2),
ideal(z-1,x1-x2,2*x0^2*y0^2*y1+3*x0*x2*y0^2*y1+2*x2^2*y0^2*y1+2*x0^2*y0*y1^2+3*x0*x2*y0*y1^2+2*x2^2*y0*y1^2+2*x0^2*y0^2*y2+3*x0*x2*y0^2*y2+2*x2^2*y0^2*y2-12*x0^2*y0*y1*y2-27*x0*x2*y0*y1*y2-12*x2^2*y0*y1*y2+2*x0^2*y1^2*y2+3*x0*x2*y1^2*y2+2*x2^2*y1^2*y2+2*x0^2*y0*y2^2+3*x0*x2*y0*y2^2+2*x2^2*y0*y2^2+2*x0^2*y1*y2^2+3*x0*x2*y1*y2^2+2*x2^2*y1*y2^2),
ideal(z-1,y2,y1,x0^2*x1+x0*x1^2+x0^2*x2-6*x0*x1*x2+x1^2*x2+x0*x2^2+x1*x2^2),
ideal(z-1,y1-y2,14*y0^2-19*y0*y2-4*y2^2,14*x0*x1*x2*y0-x0^2*x1*y2-x0*x1^2*y2-x0^2*x2*y2+x0*x1*x2*y2-x1^2*x2*y2-x0*x2^2*y2-x1*x2^2*y2,7*x0^2*x1*y0+7*x0*x1^2*y0+7*x0^2*x2*y0+7*x1^2*x2*y0+7*x0*x2^2*y0+7*x1*x2^2*y0-10*x0^2*x1*y2-10*x0*x1^2*y2-10*x0^2*x2*y2-18*x0*x1*x2*y2-10*x1^2*x2*y2-10*x0*x2^2*y2-10*x1*x2^2*y2,x0^4*x1^2+2*x0^3*x1^3+x0^2*x1^4+2*x0^4*x1*x2-19*x0^3*x1^2*x2-19*x0^2*x1^3*x2+2*x0*x1^4*x2+x0^4*x2^2-19*x0^3*x1*x2^2-30*x0^2*x1^2*x2^2-19*x0*x1^3*x2^2+x1^4*x2^2+2*x0^3*x2^3-19*x0^2*x1*x2^3-19*x0*x1^2*x2^3+2*x1^3*x2^3+x0^2*x2^4+2*x0*x1*x2^4+x1^2*x2^4),
ideal(z+1,x0-x1,2*x1^2*y0^2*y1+3*x1*x2*y0^2*y1+2*x2^2*y0^2*y1+2*x1^2*y0*y1^2+3*x1*x2*y0*y1^2+2*x2^2*y0*y1^2+2*x1^2*y0^2*y2+3*x1*x2*y0^2*y2+2*x2^2*y0^2*y2-12*x1^2*y0*y1*y2-27*x1*x2*y0*y1*y2-12*x2^2*y0*y1*y2+2*x1^2*y1^2*y2+3*x1*x2*y1^2*y2+2*x2^2*y1^2*y2+2*x1^2*y0*y2^2+3*x1*x2*y0*y2^2+2*x2^2*y0*y2^2+2*x1^2*y1*y2^2+3*x1*x2*y1*y2^2+2*x2^2*y1*y2^2),
ideal(z+1,x0-x2,2*x1^2*y0^2*y1+3*x1*x2*y0^2*y1+2*x2^2*y0^2*y1+2*x1^2*y0*y1^2+3*x1*x2*y0*y1^2+2*x2^2*y0*y1^2+2*x1^2*y0^2*y2+3*x1*x2*y0^2*y2+2*x2^2*y0^2*y2-12*x1^2*y0*y1*y2-27*x1*x2*y0*y1*y2-12*x2^2*y0*y1*y2+2*x1^2*y1^2*y2+3*x1*x2*y1^2*y2+2*x2^2*y1^2*y2+2*x1^2*y0*y2^2+3*x1*x2*y0*y2^2+2*x2^2*y0*y2^2+2*x1^2*y1*y2^2+3*x1*x2*y1*y2^2+2*x2^2*y1*y2^2),
ideal(z+1,x1-x2,2*x0^2*y0^2*y1+3*x0*x2*y0^2*y1+2*x2^2*y0^2*y1+2*x0^2*y0*y1^2+3*x0*x2*y0*y1^2+2*x2^2*y0*y1^2+2*x0^2*y0^2*y2+3*x0*x2*y0^2*y2+2*x2^2*y0^2*y2-12*x0^2*y0*y1*y2-27*x0*x2*y0*y1*y2-12*x2^2*y0*y1*y2+2*x0^2*y1^2*y2+3*x0*x2*y1^2*y2+2*x2^2*y1^2*y2+2*x0^2*y0*y2^2+3*x0*x2*y0*y2^2+2*x2^2*y0*y2^2+2*x0^2*y1*y2^2+3*x0*x2*y1*y2^2+2*x2^2*y1*y2^2),
ideal(z+1,y2,y1,x0^2*x1+x0*x1^2+x0^2*x2-6*x0*x1*x2+x1^2*x2+x0*x2^2+x1*x2^2),
ideal(z+1,y1-y2,14*y0^2-19*y0*y2-4*y2^2,14*x0*x1*x2*y0-x0^2*x1*y2-x0*x1^2*y2-x0^2*x2*y2+x0*x1*x2*y2-x1^2*x2*y2-x0*x2^2*y2-x1*x2^2*y2,7*x0^2*x1*y0+7*x0*x1^2*y0+7*x0^2*x2*y0+7*x1^2*x2*y0+7*x0*x2^2*y0+7*x1*x2^2*y0-10*x0^2*x1*y2-10*x0*x1^2*y2-10*x0^2*x2*y2-18*x0*x1*x2*y2-10*x1^2*x2*y2-10*x0*x2^2*y2-10*x1*x2^2*y2,x0^4*x1^2+2*x0^3*x1^3+x0^2*x1^4+2*x0^4*x1*x2-19*x0^3*x1^2*x2-19*x0^2*x1^3*x2+2*x0*x1^4*x2+x0^4*x2^2-19*x0^3*x1*x2^2-30*x0^2*x1^2*x2^2-19*x0*x1^3*x2^2+x1^4*x2^2+2*x0^3*x2^3-19*x0^2*x1*x2^3-19*x0*x1^2*x2^3+2*x1^3*x2^3+x0^2*x2^4+2*x0*x1*x2^4+x1^2*x2^4),
ideal(x1-x2,x0-x2,z^2-z+1,7*y0^2*y1+7*y0*y1^2+7*y0^2*y2-51*y0*y1*y2+7*y1^2*y2+7*y0*y2^2+7*y1*y2^2),
ideal(x1+x2,x0-x2,z^2-z+1,y0*y1+y0*y2+y1*y2),
ideal(y0+y1+y2,x1+x2,x0-x2,z^2-z+1),
ideal(x1-x2,x0-x2,z^2+z+1,7*y0^2*y1+7*y0*y1^2+7*y0^2*y2-51*y0*y1*y2+7*y1^2*y2+7*y0*y2^2+7*y1*y2^2),
ideal(x1+x2,x0-x2,z^2+z+1,y0*y1+y0*y2+y1*y2),
ideal(y0+y1+y2,x1+x2,x0-x2,z^2+z+1),
ideal(x1+x2,x0+x2,z^2+z+1,y0*y1+y0*y2+y1*y2),
ideal(y0+y1+y2,x1+x2,x0+x2,z^2+z+1),
ideal(x1+x2,x0+x2,z^2-z+1,y0*y1+y0*y2+y1*y2),
ideal(y0+y1+y2,x1+x2,x0+x2,z^2-z+1),
ideal(x0+x1,z^2-z+1,y0*y1+y0*y2+y1*y2,3*x1^2*y1+(4*z-2)*x1*x2*y1+3*x2^2*y1+3*x1^2*y2+(-4*z+2)*x1*x2*y2+3*x2^2*y2,3*z*x1^2*y1+(2*z-4)*x1*x2*y1+3*z*x2^2*y1+3*z*x1^2*y2+(-2*z+4)*x1*x2*y2+3*z*x2^2*y2,4*x1*x2*y0+(2*z-1)*x1^2*y2+2*x1*x2*y2+(2*z-1)*x2^2*y2,4*z*x1*x2*y0+(z-2)*x1^2*y2+2*z*x1*x2*y2+(z-2)*x2^2*y2,16*x1^2*x2^2*y0+3*x1^4*y1+10*x1^2*x2^2*y1+3*x2^4*y1+3*x1^4*y2+10*x1^2*x2^2*y2+3*x2^4*y2,3*x1^4*y1^2+10*x1^2*x2^2*y1^2+3*x2^4*y1^2+6*x1^4*y1*y2+4*x1^2*x2^2*y1*y2+6*x2^4*y1*y2+3*x1^4*y2^2+10*x1^2*x2^2*y2^2+3*x2^4*y2^2),
ideal(x1-x2,x0+x2,z^2-z+1,y0*y1+y0*y2+y1*y2),
ideal(y0+y1+y2,x1-x2,x0+x2,z^2-z+1),
ideal(x0+x1+x2,z^2-z+1,x1+(-z+1)*x2,z*x1+x2,x1^2+x1*x2+x2^2,2*y0^2*y1+2*y0*y1^2+2*y0^2*y2-3*y0*y1*y2+2*y1^2*y2+2*y0*y2^2+2*y1*y2^2),
ideal(x0+x1+x2,z^2-z+1,x1+z*x2,z*x1+(z-1)*x2,x1^2+x1*x2+x2^2,2*y0^2*y1+2*y0*y1^2+2*y0^2*y2-3*y0*y1*y2+2*y1^2*y2+2*y0*y2^2+2*y1*y2^2),
ideal(x1+x2,z^2-z+1,y0*y1+y0*y2+y1*y2,3*x0^2*y1+(4*z-2)*x0*x2*y1+3*x2^2*y1+3*x0^2*y2+(-4*z+2)*x0*x2*y2+3*x2^2*y2,3*z*x0^2*y1+(2*z-4)*x0*x2*y1+3*z*x2^2*y1+3*z*x0^2*y2+(-2*z+4)*x0*x2*y2+3*z*x2^2*y2,4*x0*x2*y0+(2*z-1)*x0^2*y2+2*x0*x2*y2+(2*z-1)*x2^2*y2,4*z*x0*x2*y0+(z-2)*x0^2*y2+2*z*x0*x2*y2+(z-2)*x2^2*y2,16*x0^2*x2^2*y0+3*x0^4*y1+10*x0^2*x2^2*y1+3*x2^4*y1+3*x0^4*y2+10*x0^2*x2^2*y2+3*x2^4*y2,3*x0^4*y1^2+10*x0^2*x2^2*y1^2+3*x2^4*y1^2+6*x0^4*y1*y2+4*x0^2*x2^2*y1*y2+6*x2^4*y1*y2+3*x0^4*y2^2+10*x0^2*x2^2*y2^2+3*x2^4*y2^2),
ideal(x0+x2,z^2-z+1,y0*y1+y0*y2+y1*y2,3*x1^2*y1+(-4*z+2)*x1*x2*y1+3*x2^2*y1+3*x1^2*y2+(4*z-2)*x1*x2*y2+3*x2^2*y2,3*z*x1^2*y1+(-2*z+4)*x1*x2*y1+3*z*x2^2*y1+3*z*x1^2*y2+(2*z-4)*x1*x2*y2+3*z*x2^2*y2,4*x1*x2*y0+(-2*z+1)*x1^2*y2+2*x1*x2*y2+(-2*z+1)*x2^2*y2,4*z*x1*x2*y0+(-z+2)*x1^2*y2+2*z*x1*x2*y2+(-z+2)*x2^2*y2,16*x1^2*x2^2*y0+3*x1^4*y1+10*x1^2*x2^2*y1+3*x2^4*y1+3*x1^4*y2+10*x1^2*x2^2*y2+3*x2^4*y2,3*x1^4*y1^2+10*x1^2*x2^2*y1^2+3*x2^4*y1^2+6*x1^4*y1*y2+4*x1^2*x2^2*y1*y2+6*x2^4*y1*y2+3*x1^4*y2^2+10*x1^2*x2^2*y2^2+3*x2^4*y2^2),
ideal(y2,y1,z^2-z+1,x0^2*x1+x0*x1^2+x0^2*x2-6*x0*x1*x2+x1^2*x2+x0*x2^2+x1*x2^2),
ideal(y1,y0+y2,z^2-z+1,x0^2*x1+z*x0*x1^2+(-z+1)*x0^2*x2+(z-1)*x1^2*x2-z*x0*x2^2-x1*x2^2,z*x0^2*x1+(z-1)*x0*x1^2+x0^2*x2-x1^2*x2+(-z+1)*x0*x2^2-z*x1*x2^2,x0^4*x1^2+x0^3*x1^3+x0^2*x1^4+x0^4*x1*x2-x0^3*x1^2*x2-x0^2*x1^3*x2+x0*x1^4*x2+x0^4*x2^2-x0^3*x1*x2^2-6*x0^2*x1^2*x2^2-x0*x1^3*x2^2+x1^4*x2^2+x0^3*x2^3-x0^2*x1*x2^3-x0*x1^2*x2^3+x1^3*x2^3+x0^2*x2^4+x0*x1*x2^4+x1^2*x2^4),
ideal(y2,y0+y1,z^2-z+1,x0^2*x1+(-z+1)*x0*x1^2+z*x0^2*x2-z*x1^2*x2+(z-1)*x0*x2^2-x1*x2^2,z*x0^2*x1+x0*x1^2+(z-1)*x0^2*x2+(-z+1)*x1^2*x2-x0*x2^2-z*x1*x2^2,x0^4*x1^2+x0^3*x1^3+x0^2*x1^4+x0^4*x1*x2-x0^3*x1^2*x2-x0^2*x1^3*x2+x0*x1^4*x2+x0^4*x2^2-x0^3*x1*x2^2-6*x0^2*x1^2*x2^2-x0*x1^3*x2^2+x1^4*x2^2+x0^3*x2^3-x0^2*x1*x2^3-x0*x1^2*x2^3+x1^3*x2^3+x0^2*x2^4+x0*x1*x2^4+x1^2*x2^4),
ideal(x0+x1,z^2+z+1,y0*y1+y0*y2+y1*y2,3*x1^2*y1+(-4*z-2)*x1*x2*y1+3*x2^2*y1+3*x1^2*y2+(4*z+2)*x1*x2*y2+3*x2^2*y2,3*z*x1^2*y1+(2*z+4)*x1*x2*y1+3*z*x2^2*y1+3*z*x1^2*y2+(-2*z-4)*x1*x2*y2+3*z*x2^2*y2,4*x1*x2*y0+(-2*z-1)*x1^2*y2+2*x1*x2*y2+(-2*z-1)*x2^2*y2,4*z*x1*x2*y0+(z+2)*x1^2*y2+2*z*x1*x2*y2+(z+2)*x2^2*y2,16*x1^2*x2^2*y0+3*x1^4*y1+10*x1^2*x2^2*y1+3*x2^4*y1+3*x1^4*y2+10*x1^2*x2^2*y2+3*x2^4*y2,3*x1^4*y1^2+10*x1^2*x2^2*y1^2+3*x2^4*y1^2+6*x1^4*y1*y2+4*x1^2*x2^2*y1*y2+6*x2^4*y1*y2+3*x1^4*y2^2+10*x1^2*x2^2*y2^2+3*x2^4*y2^2),
ideal(x1-x2,x0+x2,z^2+z+1,y0*y1+y0*y2+y1*y2),
ideal(y0+y1+y2,x1-x2,x0+x2,z^2+z+1),
ideal(x0+x1+x2,z^2+z+1,x1-z*x2,z*x1+(z+1)*x2,x1^2+x1*x2+x2^2,2*y0^2*y1+2*y0*y1^2+2*y0^2*y2-3*y0*y1*y2+2*y1^2*y2+2*y0*y2^2+2*y1*y2^2),
ideal(x0+x1+x2,z^2+z+1,x1+(z+1)*x2,z*x1-x2,x1^2+x1*x2+x2^2,2*y0^2*y1+2*y0*y1^2+2*y0^2*y2-3*y0*y1*y2+2*y1^2*y2+2*y0*y2^2+2*y1*y2^2),
ideal(x1+x2,z^2+z+1,y0*y1+y0*y2+y1*y2,3*x0^2*y1+(-4*z-2)*x0*x2*y1+3*x2^2*y1+3*x0^2*y2+(4*z+2)*x0*x2*y2+3*x2^2*y2,3*z*x0^2*y1+(2*z+4)*x0*x2*y1+3*z*x2^2*y1+3*z*x0^2*y2+(-2*z-4)*x0*x2*y2+3*z*x2^2*y2,4*x0*x2*y0+(-2*z-1)*x0^2*y2+2*x0*x2*y2+(-2*z-1)*x2^2*y2,4*z*x0*x2*y0+(z+2)*x0^2*y2+2*z*x0*x2*y2+(z+2)*x2^2*y2,16*x0^2*x2^2*y0+3*x0^4*y1+10*x0^2*x2^2*y1+3*x2^4*y1+3*x0^4*y2+10*x0^2*x2^2*y2+3*x2^4*y2,3*x0^4*y1^2+10*x0^2*x2^2*y1^2+3*x2^4*y1^2+6*x0^4*y1*y2+4*x0^2*x2^2*y1*y2+6*x2^4*y1*y2+3*x0^4*y2^2+10*x0^2*x2^2*y2^2+3*x2^4*y2^2),
ideal(x0+x2,z^2+z+1,y0*y1+y0*y2+y1*y2,3*x1^2*y1+(4*z+2)*x1*x2*y1+3*x2^2*y1+3*x1^2*y2+(-4*z-2)*x1*x2*y2+3*x2^2*y2,3*z*x1^2*y1+(-2*z-4)*x1*x2*y1+3*z*x2^2*y1+3*z*x1^2*y2+(2*z+4)*x1*x2*y2+3*z*x2^2*y2,4*x1*x2*y0+(2*z+1)*x1^2*y2+2*x1*x2*y2+(2*z+1)*x2^2*y2,4*z*x1*x2*y0+(-z-2)*x1^2*y2+2*z*x1*x2*y2+(-z-2)*x2^2*y2,16*x1^2*x2^2*y0+3*x1^4*y1+10*x1^2*x2^2*y1+3*x2^4*y1+3*x1^4*y2+10*x1^2*x2^2*y2+3*x2^4*y2,3*x1^4*y1^2+10*x1^2*x2^2*y1^2+3*x2^4*y1^2+6*x1^4*y1*y2+4*x1^2*x2^2*y1*y2+6*x2^4*y1*y2+3*x1^4*y2^2+10*x1^2*x2^2*y2^2+3*x2^4*y2^2),
ideal(y2,y1,z^2+z+1,x0^2*x1+x0*x1^2+x0^2*x2-6*x0*x1*x2+x1^2*x2+x0*x2^2+x1*x2^2),
ideal(y2,y0+y1,z^2+z+1,x0^2*x1+(z+1)*x0*x1^2-z*x0^2*x2+z*x1^2*x2+(-z-1)*x0*x2^2-x1*x2^2,z*x0^2*x1-x0*x1^2+(z+1)*x0^2*x2+(-z-1)*x1^2*x2+x0*x2^2-z*x1*x2^2,x0^4*x1^2+x0^3*x1^3+x0^2*x1^4+x0^4*x1*x2-x0^3*x1^2*x2-x0^2*x1^3*x2+x0*x1^4*x2+x0^4*x2^2-x0^3*x1*x2^2-6*x0^2*x1^2*x2^2-x0*x1^3*x2^2+x1^4*x2^2+x0^3*x2^3-x0^2*x1*x2^3-x0*x1^2*x2^3+x1^3*x2^3+x0^2*x2^4+x0*x1*x2^4+x1^2*x2^4),
ideal(y1,y0+y2,z^2+z+1,x0^2*x1-z*x0*x1^2+(z+1)*x0^2*x2+(-z-1)*x1^2*x2+z*x0*x2^2-x1*x2^2,z*x0^2*x1+(z+1)*x0*x1^2-x0^2*x2+x1^2*x2+(-z-1)*x0*x2^2-z*x1*x2^2,x0^4*x1^2+x0^3*x1^3+x0^2*x1^4+x0^4*x1*x2-x0^3*x1^2*x2-x0^2*x1^3*x2+x0*x1^4*x2+x0^4*x2^2-x0^3*x1*x2^2-6*x0^2*x1^2*x2^2-x0*x1^3*x2^2+x1^4*x2^2+x0^3*x2^3-x0^2*x1*x2^3-x0*x1^2*x2^3+x1^3*x2^3+x0^2*x2^4+x0*x1*x2^4+x1^2*x2^4)}

Hardware Overview:

Model Name: MacBook Pro Processor Name: Intel Core i7 Processor Speed: 2,7 GHz Number of Processors: 1 Total Number of Cores: 4

Macaulay2, version 1.13