|Title||Some combinatorial applications of Gr\"obner bases.|
|Author(s)|| Tamas Meszaros, Lajos Rónyai|
|Type||Book, Chapter in Book, Conference Proceeding|
|Abstract||Let 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.
|Keywords||Gröbner basis, standard monomial, lexicographic order, vanishing ideal, Hilbert function|