The lex game and some applications

Let FF be a field, VV a finite subset of FnFn. We introduce the lex game, which yields a combinatorial description of the lexicographic standard monomials of the ideal I(V)I(V) of polynomials vanishing on VV.As a consequence, we obtain a fast algorithm which computes the lexicographic standard monomials of I(V)I(V).We apply the lex game to calculate explicitly the standard monomials for special types of subsets of {0,1}n{0,1}n. For D⊆ZD⊆Z let VDVD denote the vectors y∈{0,1}n in which the number of ones (the Hamming weight of y) is in DD. We calculate the lexicographic standard monomials of VDVD, where D=D(d,ℓ,r)={a∈Z:∃a′∈Zwithd≤a′≤d+ℓ−1anda′≡a(modr)}, for d,ℓ,r∈Nd,ℓ,r∈N fixed with 0≤d