minimal Grobner basis

[Ronan Terpereau]

Hi,
I just wanted to know if the Grobner basis which is computed when I
use
G:=gens gb J
for and ideal J over a polynomial ring is always minimal in the sense
that any Grobner basis of J has at least as much generators as the
cardinal of G.
Thank you by advance,
Ronan

[Douglas Leonard]

Ronan---

I'm not the one guaranteeing anything, but I thought I would make the following comment.
I rely on gens gb to column-reduce a matrix with polynomial entries.
This always gives me a minimal, reduced answer;
but I need to know which monomial ordering is best for a certain problem and how to use the
options Position=> Up and/or Position=>Down to get the answers I wish.
This operation seems to me to be one of the big strengths of M2, and worth the price of learning it.

Doug

________________________________________
From: maca...@googlegroups.com [maca...@googlegroups.com] on behalf of Ronan Terpereau [sabre...@gmail.com]
Sent: Tuesday, February 14, 2012 6:11 AM
To: Macaulay2
Subject: [Macaulay2] minimal Grobner basis

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[Andrew Critch]

I believe the documentation does not discuss this.  But If I understand correctly from this tutorial,

http://www.math.uiuc.edu/Macaulay2/Book/ComputationsBook/chapters/varieties/chapter-wrapper.pdf

gb = grobnerBasis computes a *reduced* Groebner basis for the specified monomial ordering, which is actually unique given the ordering, and of minimal cardinality among other Groebner bases for the same ideal / module.

Actually, a reduced gb is in particular leading-term-minimal, in the sense that no leading term of any generator divides any other generator's leading term. Any leading-term-minimal Groebner basis is also cardinality-minimal, i.e. has a minimal *number* of generators.

--

Critch, Andrew
UC Berkeley
http://math.berkeley.edu/~critch/

[David Eisenbud]

BUT: different orderings, not to mention different choices of variables, lead to dramatically different Groebner bases, with different cardinalities -- so there is no reason to suppose that a given GB computed by M2 is of minimal cardinality among those!

--
David Eisenbud
Professor of Mathematics,
University of California, Berkeley
www.msri.org/~de

[Douglas Leonard]

Not to mention highlighing dramatically different structure.

Doug
________________________________________
From: maca...@googlegroups.com [maca...@googlegroups.com] on behalf of David Eisenbud [d...@msri.org]
Sent: Wednesday, February 15, 2012 6:08 AM
To: maca...@googlegroups.com
Subject: Re: [Macaulay2] minimal Grobner basis

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[Frank-Olaf Schreyer]

How does HREF works within documentation for package?

Frank.

[Daniel R. Grayson]

Does this clarify it?:

http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.4/share/doc/Macaulay2/Text/html/___H__R__E__F.html

It seems you posted your question as a followup instead of a new query, by the way.

[Florian Geiß]

I think the question was how to use it in simple doc. In this case, one writes @ HREF(…,…) @

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