Gröbner bases via linkage

E. Gorla, J. C. Migliore, U. Nagel

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8 Scopus citations

Abstract

In this paper, we give a sufficient condition for a set G of polynomials to be a Gröbner basis with respect to a given term-order for the ideal I that it generates. Our criterion depends on the linkage pattern of the ideal I and of the ideal generated by the initial terms of the elements of G. We then apply this criterion to ideals generated by minors and pfaffians. More precisely, we consider large families of ideals generated by minors or pfaffians in a matrix or a ladder, where the size of the minors or pfaffians is allowed to vary in different regions of the matrix or the ladder. We use the sufficient condition that we established to prove that the minors or pfaffians form a Gröbner basis for the ideal that they generate, with respect to any diagonal or anti-diagonal term-order. We also show that the corresponding initial ideal is Cohen-Macaulay and squarefree, and that the simplicial complex associated to it is vertex decomposable, hence shellable. Our proof relies on known results in liaison theory, combined with a simple Hilbert function computation. In particular, our arguments are completely algebraic.

Original languageEnglish
Pages (from-to)110-134
Number of pages25
JournalJournal of Algebra
Volume384
DOIs
StatePublished - Jun 5 2013

Bibliographical note

Funding Information:
E-mail addresses: elisa.gorla@unine.ch (E. Gorla), migliore.1@nd.edu (J.C. Migliore), uwe.nagel@uky.edu (U. Nagel). 1 The first author was supported by the Swiss National Science Foundation under grant No. 123393. She acknowledges financial support from the University of Notre Dame and the University of Kentucky, where part of this work was done. 2 The second author was supported by the National Security Agency under grant No. H98230-09-1-0031. 3 The third author was supported by the National Security Agency under grant No. H98230-09-1-0032.

Keywords

  • Determinantal and pfaffian ideals
  • Gröbner bases
  • Liaison

ASJC Scopus subject areas

  • Algebra and Number Theory

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