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Some Combinatorial Applications of Gröbner Bases

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Algebraic Informatics (CAI 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6742))

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Abstract

Let \(\mathbb F\) be a field, \(V \subseteq\mathbb F^n\) 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 \(\mathbb F[x_1... x_n]\). 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.

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Author information

Authors and Affiliations

  1. Computer and Automation Research Institute, Hungarian Academy of Sciences, Hungary

    Lajos Rónyai

  2. Institute of Mathematics, Budapest University of Technology and Economics, Hungary

    Lajos Rónyai

  3. Department of Mathematics, Central European University, Hungary

    Tamás Mészáros

Authors
  1. Lajos Rónyai
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  2. Tamás Mészáros
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Editors and Affiliations

  1. Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Altenberger Straße 69, 4040, Linz, Austria

    Franz Winkler

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Rónyai, L., Mészáros, T. (2011). Some Combinatorial Applications of Gröbner Bases. In: Winkler, F. (eds) Algebraic Informatics. CAI 2011. Lecture Notes in Computer Science, vol 6742. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21493-6_4

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  • DOI: https://doi.org/10.1007/978-3-642-21493-6_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-21492-9

  • Online ISBN: 978-3-642-21493-6

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