NORMALITY OF MONOMIAL IDEALS

Given the monomial ideal I = (x(1)(alpha 1), ... ,x(n)(alpha n)) subset of K[x(1), ... , x(n)], where alpha(i) are positive integers and K a field, let J be the integral closure of I. It is a challenging problem to translate the question of the normality of J into a question about the exponent set Gamma(J) and the Newtonian polyhedron NP(J). A relaxed version of this problem is to give necessary or sufficient conditions on alpha 1, ... , alpha(n) for the normality of J. We show that if alpha(i) is an element of {s,l} with s and l arbitrary positive integers, then J is normal.