Upper bounds of depth of monomial ideals

TL;DR

This paper establishes upper bounds for the depth of monomial ideals in polynomial rings by analyzing inequalities related to the counts of square-free monomials of different degrees.

Contribution

It introduces new bounds on the depth of monomial ideals based on inequalities among the counts of square-free monomials of varying degrees.

Findings

Derived inequalities provide upper bounds for depth.

Applicable to ideals generated by square-free monomials.

Enhances understanding of the algebraic structure of monomial ideals.

Abstract

Let $J\subsetneq I$ be two ideals of a polynomial ring $S$ over a field, generated by square free monomials. We show that some inequalities among the numbers of square free monomials of $I\setminus J$ of different degrees give upper bounds of $\depth_SI/J$.