Post new topic Reply to topic  [ 1 post ] 
Author Message
 Post subject: Groebner Basis and extending partial solutions
PostPosted: Sat Mar 04, 2017 3:21 pm 

Joined: Thu Feb 16, 2017 9:52 am
Posts: 1
Suppose that a finite set of polynomials in C[x,y,z] has a finite number of solutions (i.e. the generated ideal is 0-dimensional).
Suppose also that the Groebner basis with respect to lex order x>y>z is

[f(z), g(y,z), h(y,z), k(x,y,z)]

As well known, the system can be now easily solved: choose a root z0 of f, plug it into g and h and look for a common root (y0) etc.

The question is the following: Is it true that for EVERY root z0 of f there exist y0, z0 such that (x0,y0,z0) satisfy the system?

In all the examples I have seen this is true, but I don't know whether this is true in general or there is a counterexample.

Note that an extension from (y0,z0) to (x0,y0,z0) is not always possible (there is an "Extension theorem" which must be used).
The problem here is to extend from (z0) to (x0,y0,z0) which seems to be always possible. Is it?

Report this post
Reply with quote  
Display posts from previous:  Sort by  
Post new topic Reply to topic  [ 1 post ] 

You can post new topics in this forum
You can reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum
You cannot post attachments in this forum

It is currently Tue Oct 17, 2017 9:54 am
Powered by phpBB © 2000, 2002, 2005, 2007 phpBB Group