Stanley's conjecture for critical ideals

TL;DR

This paper investigates Stanley's conjecture for critical monomial ideals, providing calculations of Stanley depth and establishing results for non-critical cases, advancing understanding of Stanley depth in polynomial rings.

Contribution

It proves Stanley's conjecture for modules I and S/I when I is a critical monomial ideal and computes the Stanley depth for canonical critical monomial ideals.

Findings

Stanley's conjecture holds for critical monomial ideals

Calculated Stanley depth for canonical critical monomial ideals

Existence of Stanley ideals with same depth and Hilbert function for non-critical monomials

Abstract

Let S=K[x_1,x_2,...,x_n] be a polynomial ring in n variables over a field K. Stanley's conjecture holds for the modules I and S/I, when I is a critical monomial ideal. We calculate the Stanley depth of S/I when I is a canonical critical monomial ideal. For non critical monomial ideals we show the existence of a Stanley ideal with the same depth and Hilbert function.