Monomial localizations and polymatroidal ideals

TL;DR

This paper explores the relationship between monomial localizations and polymatroidal ideals, proposing a conjecture and proving it in specific cases, while introducing componentwise polymatroidal ideals.

Contribution

It conjectures a characterization of polymatroidal ideals via monomial localizations and extends results to a new class called componentwise polymatroidal ideals.

Findings

Proved the conjecture for squarefree monomial ideals.

Extended properties of polymatroidal ideals to componentwise polymatroidal ideals.

Established equivalence with matroid characterizations in special cases.

Abstract

In this paper we consider monomial localizations of monomial ideals and conjecture that a monomial ideal is polymatroidal if and only if all its monomial localizations have a linear resolution. The conjecture is proved for squarefree monomial ideals where it is equivalent to a well-known characterization of matroids. We prove our conjecture in many other special cases. We also introduce the concept of componentwise polymatroidal ideals and extend several of the results, known for polymatroidal ideals, to this new class of ideals.