Normality of Monomial Ideals

Abstract

Given the monomial ideal I=(x_1^{{\alpha}_1},...,x_{n}^{{\alpha}_{n}})\subset K[x_1,...,x_{n}] where {\alpha}_{i} are positive integers and K a field and 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 Newton 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}\epsilon{s,l} with s and l arbitrary positive integers, then J is normal.