Stanley Decompositions and Polarization

TL;DR

This paper explores the properties of Stanley ideals, demonstrating that polarization preserves the Stanley ideal property for Cohen-Macaulay Stanley ideals, through the introduction of nice partitions of multicomplexes.

Contribution

It introduces the concept of nice partitions of multicomplexes and proves that polarization maintains the Stanley ideal property for Cohen-Macaulay Stanley ideals.

Findings

Polarization of Cohen-Macaulay Stanley ideals results in Stanley ideals.

Nice partitions of multicomplexes are key to understanding Stanley ideals.

The main theorem links polarization with the preservation of the Stanley property.

Abstract

We define nice partitions of the multicomplex associated to a Stanley ideal. As the main result we show that if the monomial ideal $I$ is a CM Stanley ideal, then $I^p$ is a Stanley ideal as well, where $I^p$ is the polarization of $I$.