Depth and Stanley Depth of the Canonical Form of a factor of monomial ideals

TL;DR

This paper demonstrates that the depth and Stanley depth of monomial ideal factors are invariant under canonical form transformations, simplifying computations and supporting the Stanley Conjecture.

Contribution

It introduces the concept of canonical form invariance for depth and Stanley depth, and provides an algorithm to efficiently compute these invariants.

Findings

Depth and Stanley depth are invariant under canonical form.

The Stanley Conjecture holds for an ideal factor if and only if it holds for its canonical form.

The proposed algorithm significantly reduces runtime for Stanley depth computations.

Abstract

In this paper we show that the depth and the Stanley depth of the factor of two monomial ideals is invariant under taking a so called canonical form. It follows easily that the Stanley Conjecture holds for the factor if and only if it holds for its canonical form. In particular, we construct an algorithm which simplifies the depth computation and using the canonical form we massively reduce the run time for the sdepth computation.