Some combinatorial applications of Gröbner bases

Authors:
Lajos Rónyai;Tamás Mészáros
Affiliations:
Computer and Automation Research Institute, Hungarian Academy of Sciences and Institute of Mathematics, Budapest University of Technology and Economics;Department of Mathematics, Central European University
Venue:
CAI'11 Proceedings of the 4th international conference on Algebraic informatics
Year:
2011

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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.