Monomial ideals of minimal depth and trivial modifications

TL;DR

This paper investigates monomial ideals with minimal depth in polynomial rings, confirming Stanley's conjecture for these cases and demonstrating its preservation under trivial modifications.

Contribution

It proves Stanley's conjecture for classes of monomial ideals with minimal depth and shows the conjecture's stability under trivial modifications.

Findings

Stanley's conjecture holds for monomial ideals of minimal depth.

The conjecture is preserved under trivial modifications of square-free monomial ideals.

The paper provides conditions under which Stanley's conjecture is valid for these ideals.

Abstract

Let $S$ be a polynomial algebra over a field. We study classes of monomial ideals (as for example lexsegment ideals) of $S$ having minimal depth. In particular, Stanley's conjecture holds for these ideals. Also we show that if Stanley's conjecture holds for a square free monomial ideal then it holds for all its trivial modifications.