Size and Stanley depth of monomial ideals

TL;DR

This paper clarifies the relationship between the Lyubeznik size and Stanley depth of monomial ideals, resolving a previously identified gap in the proof of their connection.

Contribution

It provides a corrected proof establishing the Lyubeznik size as a lower bound for the Stanley depth of monomial ideals.

Findings

Confirmed Lyubeznik size as a lower bound for Stanley depth

Resolved a gap in the previous proof by Herzog-Popescu-Vladoiu

Strengthened understanding of monomial ideal invariants

Abstract

The Lyubeznik size of a monomial ideal $I$ of a polynomial ring $S$ is a lower bound for the Stanley depth of $I$ decreased by $1$. A proof given by Herzog-Popescu-Vladoiu had a gap which is solved here.