Normality of Monomial Ideals

TL;DR

This paper investigates the conditions under which monomial ideals are normal, focusing on the relationship between their exponent sets, Newton polyhedra, and integral closures, and provides specific criteria for normality.

Contribution

It establishes that monomial ideals generated by powers with exponents in certain sets are normal, offering new criteria for normality based on exponent conditions.

Findings

If exponents are in specific sets, the ideal is normal.

Provides criteria linking exponent sets to normality.

Advances understanding of the algebraic structure 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.