[GAP Forum] A question on fields:

[Miguel Angel Borges-Trenard]

Dear GAP Forum,

I wrote the letter below to Prof. Joyner and he suggested me to send it to the forum.

Borges.

----- Original Message ----- 
From: Miguel Angel Borges-Trenard 
To: David Joyner 
Sent: Thursday, November 09, 2006 9:13 AM
Subject: On "Re: [GAP Forum] A question on fields": A personal communication. 


Dear Professor Joyner,

I believe the key point is that there are two problems around:

Given a field k, indeterminates x1,...,xn, and polynomials g1,..., gm, p in k[x1,...,xn], decide whether:
(1) p in Ideal(g1,...,gm) (membership ideal problem). 
(2) p in k(g1,...,gm) (i.e. p is in the field of rational functions generated by g1,..., gm over k. This is  
      the field membership problem).

Problem 2 could also be formulated and solved when g1,..., gm, p are rational functions.

I believe your proposal (see below) solves Problem 1. (I think it is not necessary to compute the ideal J generated by x-y,x+y,p; even in GAP could there be alternatives of computing reduced forms, applied to p.)

But Problem 2 is more involved, I have seen a solution in the reference below (algorithms 1.8 and 1.10 there). I don't know whether there are some new and more efficient methods and I have not had time to see whether those algorithms can be implemented in GAP, but I believe it can be possible. (At first glance, it seems to be that it is necessary to compute Groebner bases over the field
k(g1,...,gm).)

With best regards,   Borges.

Miguel Angel Borges-Trenard
Departamento de Matemática
Facultad de Matemática Computación
Universidad de Oriente
Cuba

Basic Algorithms for Rational Function Fields. J. Muller-Quade and R. Steinwandt.
J. Symbolic Computation, 1999, V. 27, p. 143-170.

----- Original Message ----- 
From: Nicola Sottocornola <Nicola.Sottocornola at fastwebnet.it>
To: David Joyner <wdjoyner at gmail.com>
Cc: GAP Forum <forum at gap-system.org>
Sent: Wednesday, November 08, 2006 8:02 AM
Subject: Re: [GAP Forum] A question on fields


> Hi,
> 
> thank you for your answer. I don't know what a Groebner basis is so I  
> can't understand your idea. I just tested it with an example:
> 
> let F = Ideal(xyz, yz) and G = Ideal(xyz, yz, x).
> 
> I have found that the GB of F is {zy} and the GB of G is {zy,x}. The  
> basis are different but x is in the field generated by xyz and yz (?)
> 
> Maybe I made a misteke in computing the basis...
> 
> Nicola
> 
> 
> On 07/nov/06, at 04:37, David Joyner wrote:
> 
> > I wonder if the following idea might work:
> > You could create the ideal I generated by x-y,x+y.
> > Now create the ideal J generated by x-y,x+y,p
> > Compute the Grobner bases of I,J and compare them.
> > If the GB of I equals the GB of J then p is in F.