Regularity of powers of quadratic sequences with applications to binomial ideals

TL;DR

This paper establishes upper bounds for the regularity of powers of quadratic sequence ideals and binomial edge ideals, providing explicit calculations for specific graph classes and extending previous theoretical results.

Contribution

It generalizes existing bounds on regularity, computes explicit regularity values for certain binomial edge ideals, and extends theoretical understanding of these algebraic structures.

Findings

Upper bounds for regularity of powers of quadratic sequence ideals.

Explicit regularity calculations for binomial edge ideals of specific graphs.

Generalization of bounds for almost complete intersection and parity binomial edge ideals.

Abstract

In this article, we obtain an upper bound for the Castelnuovo-Mumford regularity of powers of an ideal generated by a homogeneous quadratic sequence in a polynomial ring in terms of the regularity of its related ideals and degrees of its generators. As a consequence, we compute upper bounds for the regularity of powers of several binomial ideals. We generalize a result of Matsuda and Murai to show that the regularity of $J^s_G$ is bounded below by $2s+\ell(G)-1$ for all $s \geq 1$, where $J_G$ denotes the binomial edge ideal of a graph $G$ and $\ell(G)$ is the length of a longest induced path in $G$. We compute the regularity of powers of binomial edge ideals of cycle graphs, star graphs, and balloon graphs explicitly. Also, we give sharp bounds for the regularity of powers of almost complete intersection binomial edge ideals and parity binomial edge ideals.