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Re: Minimum input for GroebnerBasis

  • To: mathgroup at smc.vnet.net
  • Subject: [mg90421] Re: [mg90388] Minimum input for GroebnerBasis
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Wed, 9 Jul 2008 04:51:49 -0400 (EDT)
  • References: <200807080625.CAA18049@smc.vnet.net>

TuesdayShopping wrote:
> Given a finite set of polynomials P in variables belonging to V, we compute the GroebnerBasis G. What is the set of polynomials Q (Q is a subset of P), such that (a) Q produces the same GroebnerBasis G; and, (b) if any element from the set Q is removed, Q will no longer produce G. In other words, Q is the minimum set of polynomials (from P) required, in order to produce the Groebner Basis G. If there are several Q's possible, we will want the one with the smallest number of polynomials in in it. If there are several with the same number, will want the first one we encounter. Question is how do we find Q in Mathematica?
> 
> For example, P can be {x + y + 1, 2 x + 2 y + 2, x - y}
> 
> GroebnerBasis[{x + y + 1, 2 x + 2 y + 2, x - y}, {x,y}] is {1 + 2 y, 1 + 2 x}
> 
> Q is {x + y + 1, x - y}

The only way I know to find such subset is to systematically remove one 
polynomial from P (say you remove the i-th, then call the resulting set 
P_i) and check whether the Groebner basis of P_i is the same as that of 
P (using some fixed term ordering). For each subset P_i where this 
works, try removing a second polynomial, repeat the checks, etc.

Daniel Lichtblau
Wolfram Research



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