Polarizations of powers of graded maximal ideals

TL;DR

This paper provides a comprehensive combinatorial classification of all polarizations of powers of graded maximal ideals in polynomial rings, exploring their duals and topological properties.

Contribution

It offers a complete combinatorial characterization of polarizations of powers of graded maximal ideals and describes their Alexander duals, with visualizations for specific cases.

Findings

Every polarization defines a shellable simplicial ball.

Descriptions are visualizable in three-variable and power-two cases.

Conjectures relate polarizations to topological and algebraic geometric properties.

Abstract

We give a complete combinatorial characterization of all possible polarizations of powers of the graded maximal ideal $(x_1,x_2,\cdots,x_m)^n$ of a polynomial ring in $m$ variables. We also give a combinatorial description of the Alexander duals of such polarizations. In the three variable case $m=3$ and also in the power two case $n=2$ the descriptions are easily visualized and we show that every polarization defines a (shellable) simplicial ball. We give conjectures relating to topological properties and to algebraic geometry, in particular that any polarization of an Artinian monomial ideal defines a simplicial ball.