Some combinatorial applications of Gröbner bases

L. Rónyai, Tamás Mészáros

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

8 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 publicationLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Pages65-83
Number of pages19
Volume6742 LNCS
DOIs
Publication statusPublished - 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)03029743
ISSN (Electronic)16113349

Other

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

Fingerprint

Polynomial ring
Polynomial function
Set of points
Fast Algorithm
Finite Set
Polynomials
Context
Standards
Family

Keywords

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

ASJC Scopus subject areas

  • Computer Science(all)
  • Theoretical Computer Science

Cite this

Rónyai, L., & Mészáros, T. (2011). Some combinatorial applications of Gröbner bases. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 6742 LNCS, 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

Some combinatorial applications of Gröbner bases. / Rónyai, L.; Mészáros, Tamás.

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 6742 LNCS 2011. p. 65-83 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6742 LNCS).

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

Rónyai, L & Mészáros, T 2011, Some combinatorial applications of Gröbner bases. in Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). vol. 6742 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 6742 LNCS, pp. 65-83, 4th International Conference on Algebraic Informatics, CAI 2011, Linz, Austria, 6/21/11. https://doi.org/10.1007/978-3-642-21493-6_4
Rónyai L, Mészáros T. Some combinatorial applications of Gröbner bases. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 6742 LNCS. 2011. p. 65-83. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). https://doi.org/10.1007/978-3-642-21493-6_4
Rónyai, L. ; Mészáros, Tamás. / Some combinatorial applications of Gröbner bases. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). Vol. 6742 LNCS 2011. pp. 65-83 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).
@inproceedings{213cd6a6de334ff28df7be2db9b57d65,
title = "Some combinatorial applications of Gr{\"o}bner bases",
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]. Gr{\"o}bner 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.",
keywords = "combinatorial Nullstellensatz, Gr{\"o}bner basis, Hilbert function, inclusion matrix, lexicographic order, rank formula, S-extremal set family, standard monomial, vanishing ideal",
author = "L. R{\'o}nyai and Tam{\'a}s M{\'e}sz{\'a}ros",
year = "2011",
doi = "10.1007/978-3-642-21493-6_4",
language = "English",
isbn = "9783642214929",
volume = "6742 LNCS",
series = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",
pages = "65--83",
booktitle = "Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)",

}

TY - GEN

T1 - Some combinatorial applications of Gröbner bases

AU - Rónyai, L.

AU - Mészáros, Tamás

PY - 2011

Y1 - 2011

N2 - 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.

AB - 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.

KW - combinatorial Nullstellensatz

KW - Gröbner basis

KW - Hilbert function

KW - inclusion matrix

KW - lexicographic order

KW - rank formula

KW - S-extremal set family

KW - standard monomial

KW - vanishing ideal

UR - http://www.scopus.com/inward/record.url?scp=79959993101&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79959993101&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-21493-6_4

DO - 10.1007/978-3-642-21493-6_4

M3 - Conference contribution

AN - SCOPUS:79959993101

SN - 9783642214929

VL - 6742 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 65

EP - 83

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

ER -