Polarization and spreading of monomial ideals

TL;DR

This paper characterizes monomial ideals whose polarization and iterated squarefree transformations are isomorphic via variable permutation, providing methods for construction and comparing their depth and sdepth properties.

Contribution

It introduces a characterization of monomial ideals with isomorphic polarization and squarefree transformations, along with construction methods and depth comparisons.

Findings

Identifies conditions for isomorphism between polarization and squarefree transformations.

Provides methods to construct such monomial ideals.

Analyzes and compares depth and sdepth of original and transformed ideals.

Abstract

We characterize the monomial ideals $I\subset K[x_1,\ldots,x_n]$ with the property that the polarization $I^p$ and $I^{\sigma^n}:=$ the ideal obtained from $I$ by the $n$-th iterated squarefree operator $\sigma$ are isomorphic via a permutation of variables. We give several methods to construct such ideals. We also compare the depth and sdepth of $I$ and $I^{\sigma^n}$.