# Do Groebner bases give the smallest generating set for Ideals?

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.

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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

## 1 Answer

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.

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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
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