Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Given a Reduced Groebner Basis $(f_1,\ldots,f_n)$ for an ideal $I$, can there be another basis $(g_1,\ldots,g_m)$ for $I$ where $m<n$?

I've been reading through Cox, but can't seem to find an answer.

share|improve this question
2  
Is the ideal $(x z - y^2, x^2 y - z^2, x^3 - y z)$ a counterexample? –  Zhen Lin Apr 13 '13 at 16:24
add comment

1 Answer 1

up vote 4 down vote accepted

No, Zhen has suggested a counterexample and there is another here. In practice some ideals have grobner bases that are so large we do not know what they are because the computer calculation takes too long and requires to much memory to actually complete. As far as finding a minimal generating set of an ideal choosing a grobner basis, even a reduced one, is a very bad strategy.

See also this math overflow post.

share|improve this answer
    
Where can I learn more about finding minimal generating set? Is there some notion of dimension analogous to linear algebra? –  Steven-Owen Apr 13 '13 at 18:42
2  
There are various notions of the dimension of a ring, some of which have to do with minimal generating sets of primary ideals. I would check Eisenbud and see what's in there. –  Jim Apr 13 '13 at 19:04
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.