A note on linear resolution and polymatroidal ideals

TL;DR

This paper investigates the conjecture that monomial ideals are polymatroidal if all their localizations have linear resolutions, confirming it in specific cases related to height, variable powers, and number of variables.

Contribution

It provides an affirmative answer to the conjecture for ideals of certain heights, containing specific powers, or involving at most four variables.

Findings

Confirmed the conjecture for height n-1 ideals

Validated the conjecture when the ideal contains many pure powers

Established the conjecture for monomial ideals in up to four variables

Abstract

Let $R=K[x_1,...,x_n]$ be the polynomial ring in $n$ variables over a field $K$ and $I$ be a monomial ideal generated in degree $d$. Bandari and Herzog conjectured that a monomial ideal $I$ is polymatroidal if and only if all its monomial localizations have a linear resolution. In this paper we give an affirmative answer to the conjecture in the following cases: $(i)$ ${\rm height}(I)=n-1$; $(ii)$ $I$ contains at least $n-3$ pure powers of the variables $x_1^d,...,x_{n-3}^d$; $(iii)$ $I$ is a monomial ideal in at most four variables.