Some Combinatorial Applications of Gröbner Bases

Let IF be a field, V ⊆ IF 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.