I have a system of 26 polynomial equations with 16 variables. According to the following sage computation, the Groebner basis is trivial (over the rational). The computation is very quick (less than 100ms), so I expect the absence of solution to be observable by hand. In order to obtain such an observation:
Question: Is there a way to print the computation of the Groebner basis step by step?
sage: R.<y0, y1, y2, y3, y4, y5, y6, y7, y8, x0, x1, x2, x3, x4, x5, x6>=PolynomialRing(QQ,16)
sage: Id=Ideal(x1 + 7/5*x3 + 7/5*x5 - 4/125,
....: x0 + x2 + 7/5*x4 + 7/5*x6 - 4/125,
....: 5*y0 + 5*y1 + 7*y2 + 7*y6 + 1/5,
....: 25*y0^2 + 25*y1^2 + 35*y2^2 + 35*y6^2 - 4/5,
....: 5*y0^3 + 5*y1^3 + 7*y2^3 + 7*y6^3 - y0^2 + 1/125,
....: 5*y0*y1^2 + 5*y1*y3^2 + 7*y2*y4^2 + 7*y6*y7^2 + 1/125,
....: 5*x1*y1 - y1^2 + 7*x3*y2 + 7*x5*y6 + 1/125,
....: 5*y0*y2^2 + 5*y1*y4^2 + 7*y2*y5^2 + 7*y6*y8^2 - x0 + 1/125,
....: 5*x0*y0 + 5*x2*y1 + 7*x4*y2 - y2^2 + 7*x6*y6 + 1/125,
....: 5*y1 + 5*y3 + 7*y4 + 7*y7 + 1/5,
....: 25*y0*y1 + 25*y1*y3 + 35*y2*y4 + 35*y6*y7 + 1/5,
....: 5*y0^2*y1 + 5*y1^2*y3 + 7*y2^2*y4 + 7*y6^2*y7 - y1^2 + 1/125,
....: 25*y1^2 + 25*y3^2 + 35*y4^2 + 35*y7^2 - 4/5,
....: 5*y1^3 + 5*y3^3 + 7*y4^3 + 7*y7^3 - x1 + 1/125,
....: 5*x1*y3 - y3^2 + 7*x3*y4 + 7*x5*y7 + 1/125,
....: 5*y1*y2^2 + 5*y3*y4^2 + 7*y4*y5^2 + 7*y7*y8^2 - x2 + 1/125,
....: 5*x0*y1 + 5*x2*y3 + 7*x4*y4 - y4^2 + 7*x6*y7 + 1/125,
....: 7*y2 + 7*y4 + 49/5*y5 + 49/5*y8 + 7/25,
....: 35*y0*y2 + 35*y1*y4 + 49*y2*y5 + 49*y6*y8 + 7/25,
....: 5*y0^2*y2 + 5*y1^2*y4 + 7*y2^2*y5 + 7*y6^2*y8 - y2^2 + 1/125,
....: 35*y1*y2 + 35*y3*y4 + 49*y4*y5 + 49*y7*y8 + 7/25,
....: 5*y1^2*y2 + 5*y3^2*y4 + 7*y4^2*y5 + 7*y7^2*y8 - x3 + 1/125,
....: 5*x1*y4 - y4^2 + 7*x3*y5 + 7*x5*y8 + 1/125,
....: 35*y2^2 + 35*y4^2 + 49*y5^2 + 49*y8^2 - 18/25,
....: 5*y2^3 + 5*y4^3 + 7*y5^3 + 7*y8^3 - x4 + 1/125,
....: 5*x0*y2 + 5*x2*y4 + 7*x4*y5 - y5^2 + 7*x6*y8 + 1/125)
sage: %time G=Id.groebner_basis()
CPU times: user 78 ms, sys: 16 ms, total: 94 ms
Wall time: 78.6 ms
sage: G
[1]
