Bounded powers of edge ideals: regularity and linear quotients

TL;DR

This paper investigates the properties of bounded powers of edge ideals in polynomial rings, establishing bounds on regularity and demonstrating that certain bounded powers are polymatroidal ideals, with implications for algebraic combinatorics.

Contribution

It introduces bounds on regularity for bounded powers of edge ideals and proves these bounded powers are polymatroidal, advancing understanding of their algebraic structure.

Findings

Bounded powers of edge ideals are polymatroidal.

Regularity of bounded powers is linearly bounded by the power.

Results apply to graph edge ideals, linking combinatorics and algebra.

Abstract

Let $S=K[x_1, \ldots,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and let $I \subset S$ be a monomial ideal. For a vector $\mathfrak{c}\in\mathbb{N}^n$, we set $I_{\mathfrak{c}}$ to be the ideal generated by monomials belonging to $I$ whose exponent vectors are componentwise bounded above by $\mathfrak{c}$. Also, let $\delta_{\mathfrak{c}}(I)$ be the largest integer $k$ such that $(I^k)_{\mathfrak{c}}\neq 0$. It is shown that for every graph $G$ with edge ideal $I(G)$, the ideal $(I(G)^{\delta_{\mathfrak{c}}(I)})_{\mathfrak{c}}$ is a polymatroidal ideal. Moreover, we show that for each integer $s=1, \ldots \delta_{\mathfrak{c}}(I(G))$, the Castelnuovo--Mumford regularity of $(I(G)^s)_{\mathfrak{c}}$ is bounded above by $\delta_{\mathfrak{c}}(I(G))+s$.