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Old 2020-10-16, 06:59   #1
wblipp
 
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"William"
May 2003
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Default 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
xi + yj2 + ykyn ...
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?
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Old 2020-10-16, 08:05   #2
Nick
 
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Dec 2012
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Quote:
Originally Posted by wblipp View Post
Is there a book...that covers topics like this?
It's not my area (and I don't know how far you are already), but the books people usually start with are:
  • Eisenbud: Commutative Algebra with a view toward algebraic geometry
  • Cox, Little & O'Shea: Ideals, varieties and algorithms

Last fiddled with by Nick on 2020-10-16 at 08:11 Reason: fixed a typo
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Old 2020-10-21, 07:23   #3
jwaltos
 
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Quote:
Originally Posted by wblipp View Post
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?
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.
https://projecteuclid.org/download/p....em/1048515872

Last fiddled with by jwaltos on 2020-10-21 at 07:35
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