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2019-09-25 01:25:46 +0200 commented answer Find solution to Polynomial Sequences without going through variety

Thanks for your response! I understand the point that you are making. I was just wondering that, since I already have a Groebner basis computed, when I call the "solve" method on it, it shouldn't try to spend time recomputing the Groebner basis itself. Maybe we can consider having a lazy option for the variety computation which before everything queries the ideal if it is a Groebner basis first. In any case, for practical purposes, finding solutions to systems of polynomial equations more efficiently is always something we should aim for (and Maple is doing a great job in that front).

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2019-08-03 20:44:03 +0200 asked a question Find solution to Polynomial Sequences without going through variety

Hi everyone, I'm computing the Groebner basis of an ideal defined over the QQ ring. Once I have this Groebner basis, I would like to obtain a set of values that satisfy the equations in the Groebner basis. I know that the full set is going to be the variety of the ideal, but since this object is huge, I might not be interested in finding those values. In Maple, after computing the Groebner basis, I'm able to call the solve() method on it and even set a maximum number of solutions I want to obtain. As a small example:

I = P.ideal(x*(x-1), y*(y-1), z*(z-1), t - x*y*z)
gb = I.groebner_basis()

Here I could have called I.variety() or gb.variety() and obtained the same set of solutions:

sage: gbI.variety()
[{y: 0, z: 0, t: 0, x: 0},
 {y: 0, z: 0, t: 0, x: 1},
 {y: 1, z: 0, t: 0, x: 0},
 {y: 1, z: 0, t: 0, x: 1},
 {y: 0, z: 1, t: 0, x: 0},
 {y: 0, z: 1, t: 0, x: 1},
 {y: 1, z: 1, t: 0, x: 0},
 {y: 1, z: 1, t: 1, x: 1}]

But I would like to know if it is possible to call a method like (I can do this in Maple):


Such that I can obtain a subset of the variety instead of the whole set. My motivation is that the size of the initial system of polynomials that I have is considerably larger than this example, and finding the feasible solutions on the reduced Groebner basis is more manageable. I might also not be interested in all the elements in the variety. Finally, if I transform the Groebner basis in an ideal itself and try to compute the variety on that object

gbI = ideal(gb)

I find the following error

RuntimeError: error in Singular function call 'groebner':
int overflow in hilb 3
error occurred in or before standard.lib::stdhilb line 299: `    intvec hi = hilb( Id(1),1,W );`
expected intvec-expression. type 'help intvec;'
leaving standard.lib::stdhilb
leaving standard.lib::groebner