I would like to compute a Groebner basis for the following polynomial system of 59 variables and 69 equations. The computation crashed on my laptop (16Go of RAM) for $p=0$ or $p=5$, because requiring too much RAM.
Question: Is there a way to compute such a Groebner basis, or is it hopeless?
I can only compute it for $p=2$:
sage: function(2)
[1]
Code
def function(p):
if p==0:
F=QQ
else:
F=GF(p)
R.<u0,u1,u2,u3,u4,u5,u6,u7,u8,u9,u10,u11,u12,u13,u14,u15,u16,u17,u18,u19,u20,u21,u22,u23,u24,u25,u26,u27,u28,v0,v1,v2,v3,v4,v5,v6,v7,v8,v9,v10,v11,v12,v13,v14,v15,v16,v17,v18,v19,v20,v21,v22,v23,v24,v25,v26,v27,v28,v29>=PolynomialRing(F,59)
Eq=[u0+11/F(9)*u13+11/F(9)*u17+11/F(9)*u21+11/F(9)*u25+u3+u6-8/F(729),
u1+u10+11/F(9)*u14+11/F(9)*u18+11/F(9)*u22+11/F(9)*u26+u7-8/F(729),
u11+11/F(9)*u15+11/F(9)*u19+11/F(9)*u23+11/F(9)*u27+u4+u8-8/F(729),
u12+11/F(9)*u16+u2+11/F(9)*u20+11/F(9)*u24+11/F(9)*u28+u5+u9-10/F(891),
9*v0+9*v1+11*v15+9*v2+11*v20+11*v25+9*v3+11*v4+1/F(9),
81*v0^2+81*v1^2+99*v15^2+81*v2^2+99*v20^2+99*v25^2+81*v3^2+99*v4^2-8/F(9),
9*v0^3+9*v1^3+11*v15^3+9*v2^3+11*v20^3+11*v25^3+9*v3^3+11*v4^3-v0^2+1/F(729),
9*v0*v1^2+11*v15*v16^2+11*v20*v21^2+11*v25*v26^2+9*v1*v5^2+9*v2*v6^2+9*v3*v7^2+11*v4*v8^2-u0+1/F(729),
9*u0*v0+9*u3*v1-v1^2+11*u17*v15+9*u6*v2+11*u21*v20+11*u25*v25+11*u13*v4+1/F(729),
11*v15*v17^2+9*v0*v2^2+11*v20*v22^2+11*v25*v27^2+9*v10^2*v3+11*v11^2*v4+9*v1*v6^2+9*v2*v9^2-u1+1/F(729),
9*u1*v0+11*u18*v15+9*u7*v2-v2^2+11*u22*v20+11*u26*v25+9*u10*v3+11*u14*v4+1/F(729),
11*v15*v18^2+9*v10^2*v2+11*v20*v23^2+11*v25*v28^2+9*v12^2*v3+9*v0*v3^2+11*v13^2*v4+9*v1*v7^2+1/F(729),
9*u4*v1+11*u19*v15+9*u8*v2+11*u23*v20+11*u27*v25+9*u11*v3-v3^2+11*u15*v4+1/F(729),
11*v15*v19^2+9*v11^2*v2+11*v20*v24^2+11*v25*v29^2+9*v13^2*v3+11*v14^2*v4+9*v0*v4^2+9*v1*v8^2-u2+1/F(729),
9*u2*v0+9*u5*v1+11*u20*v15+9*u9*v2+11*u24*v20+11*u28*v25+9*u12*v3+11*u16*v4-v4^2+1/F(891),
9*v1+11*v16+11*v21+11*v26+9*v5+9*v6+9*v7+11*v8+1/F(9),
81*v0*v1+99*v15*v16+99*v20*v21+99*v25*v26+81*v1*v5+81*v2*v6+81*v3*v7+99*v4*v8+1/F(9),
9*v0^2*v1+11*v15^2*v16+11*v20^2*v21+11*v25^2*v26+9*v1^2*v5+9*v2^2*v6+9*v3^2*v7+11*v4^2*v8-v1^2+1/F(729),
81*v1^2+99*v16^2+99*v21^2+99*v26^2+81*v5^2+81*v6^2+81*v7^2+99*v8^2-8/F(9),
9*v1^3+11*v16^3+11*v21^3+11*v26^3+9*v5^3+9*v6^3+9*v7^3+11*v8^3-u3+1/F(729),
9*u0*v1+11*u17*v16+11*u21*v21+11*u25*v26+9*u3*v5-v5^2+9*u6*v6+11*u13*v8+1/F(729),
11*v16*v17^2+9*v1*v2^2+11*v21*v22^2+11*v26*v27^2+9*v5*v6^2+9*v10^2*v7+11*v11^2*v8+9*v6*v9^2+1/F(729),
9*u1*v1+11*u18*v16+11*u22*v21+11*u26*v26+9*u7*v6-v6^2+9*u10*v7+11*u14*v8+1/F(729),
11*v16*v18^2+11*v21*v23^2+11*v26*v28^2+9*v1*v3^2+9*v10^2*v6+9*v12^2*v7+9*v5*v7^2+11*v13^2*v8-u4+1/F(729),
11*u19*v16+11*u23*v21+11*u27*v26+9*u4*v5+9*u8*v6+9*u11*v7-v7^2+11*u15*v8+1/F(729),
11*v16*v19^2+11*v21*v24^2+11*v26*v29^2+9*v1*v4^2+9*v11^2*v6+9*v13^2*v7+11*v14^2*v8+9*v5*v8^2-u5+1/F(729),
9*u2*v1+11*u20*v16+11*u24*v21+11*u28*v26+9*u5*v5+9*u9*v6+9*u12*v7+11*u16*v8-v8^2+1/F(891),
9*v10+11*v11+11*v17+9*v2+11*v22+11*v27+9*v6+9*v9+1/F(9),
99*v15*v17+81*v0*v2+99*v20*v22+99*v25*v27+81*v10*v3+99*v11*v4+81*v1*v6+81*v2*v9+1/F(9),
11*v15^2*v17+9*v0^2*v2+11*v20^2*v22+11*v25^2*v27+9*v10*v3^2+11*v11*v4^2+9*v1^2*v6+9*v2^2*v9-v2^2+1/F(729),
99*v16*v17+81*v1*v2+99*v21*v22+99*v26*v27+81*v5*v6+81*v10*v7+99*v11*v8+81*v6*v9+1/F(9),
11*v16^2*v17+9*v1^2*v2+11*v21^2*v22+11*v26^2*v27+9*v5^2*v6+9*v10*v7^2+11*v11*v8^2+9*v6^2*v9-u6+1/F(729),
11*u13*v11+11*u17*v17+9*u0*v2+11*u21*v22+11*u25*v27+9*u3*v6-v6^2+9*u6*v9+1/F(729),
81*v10^2+99*v11^2+99*v17^2+81*v2^2+99*v22^2+99*v27^2+81*v6^2+81*v9^2-8/F(9),
9*v10^3+11*v11^3+11*v17^3+9*v2^3+11*v22^3+11*v27^3+9*v6^3+9*v9^3-u7+1/F(729),
9*u10*v10+11*u14*v11+11*u18*v17+9*u1*v2+11*u22*v22+11*u26*v27+9*u7*v9-v9^2+1/F(729),
9*v10*v12^2+11*v11*v13^2+11*v17*v18^2+11*v22*v23^2+11*v27*v28^2+9*v2*v3^2+9*v6*v7^2+9*v10^2*v9-u8+1/F(729),
9*u11*v10-v10^2+11*u15*v11+11*u19*v17+11*u23*v22+11*u27*v27+9*u4*v6+9*u8*v9+1/F(729),
9*v10*v13^2+11*v11*v14^2+11*v17*v19^2+11*v22*v24^2+11*v27*v29^2+9*v2*v4^2+9*v6*v8^2+9*v11^2*v9-u9+1/F(729),
9*u12*v10+11*u16*v11-v11^2+11*u20*v17+9*u2*v2+11*u24*v22+11*u28*v27+9*u5*v6+9*u9*v9+1/F(891),
9*v10+9*v12+11*v13+11*v18+11*v23+11*v28+9*v3+9*v7+1/F(9),
99*v15*v18+81*v10*v2+99*v20*v23+99*v25*v28+81*v0*v3+81*v12*v3+99*v13*v4+81*v1*v7+1/F(9),
11*v15^2*v18+9*v10*v2^2+11*v20^2*v23+11*v25^2*v28+9*v0^2*v3+9*v12*v3^2+11*v13*v4^2+9*v1^2*v7-v3^2+1/F(729),
99*v16*v18+99*v21*v23+99*v26*v28+81*v1*v3+81*v10*v6+81*v12*v7+81*v5*v7+99*v13*v8+1/F(9),
11*v16^2*v18+11*v21^2*v23+11*v26^2*v28+9*v1^2*v3+9*v10*v6^2+9*v5^2*v7+9*v12*v7^2+11*v13*v8^2+1/F(729),
9*u6*v10+11*u13*v13+11*u17*v18+11*u21*v23+11*u25*v28+9*u0*v3+9*u3*v7-v7^2+1/F(729),
81*v10*v12+99*v11*v13+99*v17*v18+99*v22*v23+99*v27*v28+81*v2*v3+81*v6*v7+81*v10*v9+1/F(9),
9*v10^2*v12+11*v11^2*v13+11*v17^2*v18+11*v22^2*v23+11*v27^2*v28+9*v2^2*v3+9*v6^2*v7+9*v10*v9^2-u10+1/F(729),
9*u7*v10-v10^2+9*u10*v12+11*u14*v13+11*u18*v18+11*u22*v23+11*u26*v28+9*u1*v3+1/F(729),
81*v10^2+81*v12^2+99*v13^2+99*v18^2+99*v23^2+99*v28^2+81*v3^2+81*v7^2-8/F(9),
9*v10^3+9*v12^3+11*v13^3+11*v18^3+11*v23^3+11*v28^3+9*v3^3+9*v7^3-u11+1/F(729),
9*u8*v10+9*u11*v12-v12^2+11*u15*v13+11*u19*v18+11*u23*v23+11*u27*v28+9*u4*v7+1/F(729),
9*v10*v11^2+9*v12*v13^2+11*v13*v14^2+11*v18*v19^2+11*v23*v24^2+11*v28*v29^2+9*v3*v4^2+9*v7*v8^2-u12+1/F(729),
9*u9*v10+9*u12*v12+11*u16*v13-v13^2+11*u20*v18+11*u24*v23+11*u28*v28+9*u2*v3+9*u5*v7+1/F(891),
11*v11+11*v13+121/F(9)*v14+121/F(9)*v19+121/F(9)*v24+121/F(9)*v29+11*v4+11*v8+11/F(81),
121*v15*v19+99*v11*v2+121*v20*v24+121*v25*v29+99*v13*v3+99*v0*v4+121*v14*v4+99*v1*v8+11/F(81),
11*v15^2*v19+9*v11*v2^2+11*v20^2*v24+11*v25^2*v29+9*v13*v3^2+9*v0^2*v4+11*v14*v4^2+9*v1^2*v8-v4^2+1/F(729),
121*v16*v19+121*v21*v24+121*v26*v29+99*v1*v4+99*v11*v6+99*v13*v7+121*v14*v8+99*v5*v8+11/F(81),
11*v16^2*v19+11*v21^2*v24+11*v26^2*v29+9*v1^2*v4+9*v11*v6^2+9*v13*v7^2+9*v5^2*v8+11*v14*v8^2-u13+1/F(729),
9*u6*v11+11*u13*v14+11*u17*v19+11*u21*v24+11*u25*v29+9*u0*v4+9*u3*v8-v8^2+1/F(729),
99*v10*v13+121*v11*v14+121*v17*v19+121*v22*v24+121*v27*v29+99*v2*v4+99*v6*v8+99*v11*v9+11/F(81),
9*v10^2*v13+11*v11^2*v14+11*v17^2*v19+11*v22^2*v24+11*v27^2*v29+9*v2^2*v4+9*v6^2*v8+9*v11*v9^2-u14+1/F(729),
9*u7*v11-v11^2+9*u10*v13+11*u14*v14+11*u18*v19+11*u22*v24+11*u26*v29+9*u1*v4+1/F(729),
99*v10*v11+99*v12*v13+121*v13*v14+121*v18*v19+121*v23*v24+121*v28*v29+99*v3*v4+99*v7*v8+11/F(81),
9*v10^2*v11+9*v12^2*v13+11*v13^2*v14+11*v18^2*v19+11*v23^2*v24+11*v28^2*v29+9*v3^2*v4+9*v7^2*v8-u15+1/F(729),
9*u8*v11+9*u11*v13-v13^2+11*u15*v14+11*u19*v19+11*u23*v24+11*u27*v29+9*u4*v8+1/F(729),
99*v11^2+99*v13^2+121*v14^2+121*v19^2+121*v24^2+121*v29^2+99*v4^2+99*v8^2-70/F(81),
9*v11^3+9*v13^3+11*v14^3+11*v19^3+11*v24^3+11*v29^3+9*v4^3+9*v8^3-u16+1/F(729),
9*u9*v11+9*u12*v13+11*u16*v14-v14^2+11*u20*v19+11*u24*v24+11*u28*v29+9*u2*v4+9*u5*v8+1/F(891)]
Id=R.ideal(Eq)
G=Id.groebner_basis()
return G
