Some combinatorial applications of Gröbner bases

Lajos Rónyai, Tamás Mészáros

Research output: Chapter in Book/Report/Conference proceedingConference contribution

12 Citations (Scopus)

Abstract

Let IF be a field, V ⊆ IFn be a (combinatorially interesting) finite set of points. Several important properties of V are reflected by the polynomial functions on V. To study these, one often considers I(V), the vanishing ideal of V in the polynomial ring IF[x1,..., xn]. Gröbner bases and standard monomials of I(V) appear to be useful in this context, leading to structural results on V. Here we survey some work of this type. At the end of the paper a new application of this kind is presented: an algebraic characterization of shattering-extremal families and a fast algorithm to recognize them.

Original languageEnglish
Title of host publicationAlgebraic Informatics - 4th International Conference, CAI 2011, Proceedings
Pages65-83
Number of pages19
DOIs
Publication statusPublished - Jul 11 2011
Event4th International Conference on Algebraic Informatics, CAI 2011 - Linz, Austria
Duration: Jun 21 2011Jun 24 2011

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume6742 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Other

Other4th International Conference on Algebraic Informatics, CAI 2011
CountryAustria
CityLinz
Period6/21/116/24/11

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Keywords

  • Gröbner basis
  • Hilbert function
  • S-extremal set family
  • combinatorial Nullstellensatz
  • inclusion matrix
  • lexicographic order
  • rank formula
  • standard monomial
  • vanishing ideal

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

Rónyai, L., & Mészáros, T. (2011). Some combinatorial applications of Gröbner bases. In Algebraic Informatics - 4th International Conference, CAI 2011, Proceedings (pp. 65-83). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6742 LNCS). https://doi.org/10.1007/978-3-642-21493-6_4