Consider the following polynomial system:
sage: R.<a, di, p, q, v0, v1, v2, v3, v4, v5>=PolynomialRing(QQ,order='invlex')
sage: Eq
[q^7 - 1,
-q + p^2,
a*p - 1,
7*di^2 + 7*di - 1,
v0^2 + di - 1/7,
v1*v6 - 1/7*a,
v2*v5 - 1/7*a^4,
v3*v4 - 1/7*a^9,
v3*v4 - 1/7*a^2,
v2*v5 - 1/7*a^11,
-1/7*a^8 + v1*v6,
1/7*p^11*v2*v5 + 1/7*p^9*v3*v4 + 1/7*p^8*v1*v6 + 1/7*p^4*v2*v5 + 1/7*p^2*v3*v4 - v0^3 + 1/7*p*v1*v6 + 1/7*v0^2 - di,
1/7*p^11*q^2*v3*v5 + 1/7*p^9*q^4*v3*v5 + 1/7*p^8*q*v2*v6 + 1/7*p^4*q^5*v2*v6 + 1/7*p*q^6*v0*v1 + 1/7*p^2*q^3*v4^2 - v0*v1*v6 + 1/7*v0*v1,
1/7*p^11*q^4*v4*v5 + 1/7*p^9*q*v3*v6 + 1/7*p^8*q^2*v3*v6 + 1/7*p^2*q^6*v4*v5 + 1/7*p^4*q^3*v0*v2 + 1/7*p*q^5*v1^2 - v0*v2*v5 + 1/7*v0*v2,
1/7*p^11*q^6*v5^2 + 1/7*p^9*q^5*v0*v3 + 1/7*p^8*q^3*v4*v6 + 1/7*p^4*q*v1*v2 + 1/7*p*q^4*v1*v2 + 1/7*p^2*q^2*v4*v6 - v0*v3*v4 + 1/7*v0*v3,
1/7*p^11*q*v5*v6 + 1/7*p^8*q^4*v5*v6 + 1/7*p^9*q^2*v1*v3 + 1/7*p^4*q^6*v2^2 + 1/7*p^2*q^5*v0*v4 + 1/7*p*q^3*v1*v3 - v0*v3*v4 + 1/7*v0*v4,
1/7*p^9*q^6*v2*v3 + 1/7*p^11*q^3*v0*v5 + 1/7*p^8*q^5*v6^2 + 1/7*p^4*q^4*v2*v3 + 1/7*p^2*q*v1*v4 + 1/7*p*q^2*v1*v4 - v0*v2*v5 + 1/7*v0*v5,
1/7*p^11*q^5*v1*v5 + 1/7*p^8*q^6*v0*v6 + 1/7*p^9*q^3*v3^2 + 1/7*p^4*q^2*v2*v4 + 1/7*p^2*q^4*v2*v4 + 1/7*p*q*v1*v5 - v0*v1*v6 + 1/7*v0*v6]
sage: Id=Ideal(Eq)
Then it required one week to compute its Groebner basis:
sage: L=list(Id.groebner_basis())
sage: L
[v5 + v1 + 1/2*a^3*di - 1/2*a^2*di + 1/2*a*di + di - 1/6*a^3 + 1/6*a^2 + 1/3*a + 1/6,
v4 - a^2*v1 - a*v1 - a^2*di - 1/6*a^3 - 1/2*a^2 - 1/6*a,
v3 + 1/6*a^3 + 1/3*a^2 - 1/3*a - 1/6,
v2 + a^2*v1 + a*v1 + 1/2*a^3*di + 1/2*a^2*di + 1/2*a*di + 1/6*a^3 + 1/6*a,
v1^2 + 1/2*a^3*di*v1 - 1/2*a^2*di*v1 + 1/2*a*di*v1 + di*v1 - 1/6*a^3*v1 + 1/6*a^2*v1 + 1/3*a*v1 + 1/6*v1 + 1/6*a,
v0 + di,
q + a^2 - 1,
p + a^3 - a,
di^2 + di - 1/6,
a^4 - a^2 + 1]
whereas the system looks not so big.
Question: Why is it so long? Is there a way to make the computation much quicker?
