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-   -   Linearly Independent Quadratics in a 7-variable polynomial Ideal (https://www.mersenneforum.org/showthread.php?t=26088)

 wblipp 2020-10-16 06:59

Linearly Independent Quadratics in a 7-variable polynomial Ideal

I'm just beginning to learn about polynomial Ideals and groebner basis. I've got a system of polynomials in seven variables and groebner basis using lex order for several orderings of the variables. The number of quadratics in the basis varies depending on the order. This is because several of the original polynomials are of the form
x[SUB]i[/SUB] + y[SUB]j[/SUB][SUP]2[/SUP] + y[SUB]k[/SUB]y[SUB]n[/SUB] ...
so it makes a big difference if the x's or the y's come first in the variable ordering.

I'd like to get a maximal set of linearly independent quadratics for the Ideal. How should I proceed? Would gradlex or gradrevlex automatically do this? Is there some cookbook procedure like the ones used for saturation or intersection of ideals? Is there a book for applied algebraic topology that covers topics like this?

 Nick 2020-10-16 08:05

[QUOTE=wblipp;560022]Is there a book...that covers topics like this?[/QUOTE]
It's not my area (and I don't know how far you are already), but the books people usually start with are:
[LIST][*]Eisenbud: Commutative Algebra with a view toward algebraic geometry[*]Cox, Little & O'Shea: Ideals, varieties and algorithms[/LIST]

 jwaltos 2020-10-21 07:23

[QUOTE=wblipp;560022]I'm just beginning to learn about polynomial Ideals and groebner basis...Is there a book for applied algebraic topology that covers topics like this?[/QUOTE]

Scilab and some other available software (GAP, Singular..) have such functions available.
There are also some worksheets in other CAS's that are available which may provide some perspective.
Grobner Bases, Springer-Verlag..Thomas Becker. (1993), is a good a starting place as any.
I would suggest you look at Buchberger's algorthm first and fit the pieces you have into place.
This is an interesting topic..solvable and unsolvable Thue equations is one arena you may wish to enter and tangle with a few equations for some sport and here are two references:
The Algorithmic Resolution of Diophantine Equations by Nigel Smart.