Depth and minimal number of generators of square free monomial ideals

TL;DR

This paper investigates the depth of square-free monomial ideals in polynomial rings, establishing conditions under which the depth is bounded above by a certain degree, thereby supporting Stanley's Conjecture.

Contribution

It provides a new criterion relating the number of generators of a certain degree to the depth of the ideal, confirming Stanley's Conjecture in these cases.

Findings

Depth is at most d under specified conditions.

Supports Stanley's Conjecture for these ideals.

Relates the number of degree d generators to the ideal's depth.

Abstract

Let $I$ be an ideal of a polynomial algebra $S$ over a field generated by square free monomials of degree $\geq d$. If $I$ contains more monomials of degree $d$ than $(n-d)/(n-d+1)$ of the total number of square free monomials of $S$ of degree $d+1$ then $\depth_SI\leq d$, in particular the Stanley's Conjecture holds in this case.