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| solutions to polynomial system avoiding some "trivial" ones https://www.singular.uni-kl.de/forum/viewtopic.php?f=10&t=2960 |
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| Author: | laurentb [ Fri Sep 10, 2021 3:40 pm ] |
| Post subject: | solutions to polynomial system avoiding some "trivial" ones |
I'm completely new to Singular, and wish to obtain "non-trivial" solutions to a system of equations. More precisely, I create a ring Code: ring r = 0,(u(1..5),v(1..5),x(1..5),y(1..5)),lp; and an ideal Code: ideal i = ...; and its Gröbner basis. I want to find u(...),v(...),x(...),y(...) that satisfy the equations in i. Two problems: (1) i has positive dimension (actually projective of dimension 11, degree 25), so Code: LIB "solve.lib"; solve(i); does not work(2) there are some trivial solutions, for example u=x,v=y which I want to avoid I tried to specialize some variables at random (by defining ideal j = i,u(1)-1,...), but didn't manage to find a non-trivial 0-dimensional ideal in this manner, so something more systematic is welcome! |
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| Author: | leonada [ Sun Oct 10, 2021 9:12 pm ] |
| Post subject: | Re: solutions to polynomial system avoiding some "trivial" ones |
Avoiding trivialities can be done by using saturation, so relative to ideals such as ideal j=u-v. This can be found in Singular. Finding varieties that are not 0-dimensional from a lex GB is more problematic. These are usually not implemented. Look up elimination and extension on the web. I wrote Macaulay2 code for this in a 2019 paper (on arXiv) using multi-homogeneous coordinates rather than projective ones, as I thought this did what was in Cox, Little, O'Shea without having to deal with exceptions. But I'm sure there are other papers on the subject as well. |
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