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TitleSome combinatorial applications of Gr\"obner bases.
Author(s) Tamas Meszaros, Lajos Rónyai
TypeBook, Chapter in Book, Conference Proceeding
AbstractLet 𝔽 be a field, V⊆𝔽n 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 𝔽[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.
KeywordsGröbner basis, standard monomial, lexicographic order, vanishing ideal, Hilbert function
URL http://link.springer.com/chapter/10.1007%2F978-3-642-21493-6_4
PublisherBerlin: Springer
Translation No
Refereed No