A theorem on refining division orders by the reverse lexicographic order
type: Item
instance of: scholarly article
title: A theorem on refining division orders by the reverse lexicographic order
zbMATH Open document ID: 0638.06003
DOI: 10.1215/S0012-7094-87-05517-7
author: Dave Bayer
author: Mike Stillman
published in: Duke Mathematical Journal
publication date: 1987-01-01T00:00:00Z
review text: If \(S_ d\) is the K-vector space of homogeneous polynomials of degree d in \(S=k[x_ 1,...,x_ n]\), where K is an infinite field, then a (partial) order \(>\) on the monomials of \(S_ d\) for each d is compatible with the monoid structure of the monomials of S if \(x^ A>x^ B\) implies \(x^ Cx^ A>x^ Cx^ B\) for all monomials \(x^ C\). If the order \(>\) is strict, then it can be used in applying the division algorithm to construct a standard basis for homogeneous ideals I whose properties vary with I. Many computaions of this type lead to results which are quite interesting from the combinatorial viewpoint as is illustrated by the algebraic theory of shellable posets, for example. In this paper the interest is more directly algebro-geometric. In particular the main theorem (theorem 6) is as follows: If \(>_ 3\) is the refinement of an (arbitrary) order \(>_ 1\) by the reverse lexicographic order \(>\) (rlex), and if \(I\subset S\) is a homogeneous ideal, then there exists a Zariski open subset \(U\subset GL(n)\) such that for any \(g\in U\), \(in_ 1(g\cdot U)\) and \(in_ 3(g\cdot I)\) have the same regularity. As noted, the reverse lexicographic order is often a most efficient choice for constructing standard bases when one can make a free choice of order, as in the computation of syzygies or Hilbert functions. These latter computations have important consequences in combinatorics also. Theorem 6 carries some of these efficiency properties of reverse lexicographic order to more general settings, which will be useful in a variety of applications.
Mathematics Subject Classification ID: 06A06
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 05A99
Mathematics Subject Classification ID: 13-04
zbMATH DE Number: 4039965
zbMATH Keywords: refinement
zbMATH Keywords: vector space of homogeneous polynomials
zbMATH Keywords: monomials
zbMATH Keywords: monoid structure
zbMATH Keywords: division algorithm
zbMATH Keywords: standard basis for homogeneous ideals
zbMATH Keywords: reverse lexicographic order
reviewed by: J. Neggers
MaRDI profile type: Publication
cites work: Uniformizing functions for certain Shimura curves, in the case D=6
cites work: Standard bases and geometric invariant theory. I: Initial ideals and state polytopes
cites work: A criterion for detecting m-regularity
cites work: Q4050885
cites work: New constructive methods in classical ideal theory
cites work: Über B. Buchbergers Verfahren, Systeme algebraischer Gleichungen zu lösen
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description: scientific article; zbMATH DE number 4039965