The lex game and some applications

Let be a field, V a finite subset of . We introduce the lex game, which yields a combinatorial description of the lexicographic standard monomials of the ideal I(V) of polynomials vanishing on V. As a consequence, we obtain a fast algorithm which computes the lexicographic standard monomials of I(V). We apply the lex game to calculate explicitly the standard monomials for special types of subsets of {0,1}n. For let VD denote the vectors in which the number of ones (the Hamming weight of ) is in D. We calculate the lexicographic standard monomials of VD, where for fixed with 0≤d