New bounds on Castelnuovo--Mumford regularity of monomial curves and application to sumsets

TL;DR

This paper introduces improved bounds on the Castelnuovo--Mumford regularity of monomial curves using Apery sets and Frobenius numbers, with applications to sumsets and algebraic properties.

Contribution

It provides new bounds on regularity under specific conditions, surpassing Lvovsky's bound, and offers algorithms to determine algebraic properties of monomial curves.

Findings

New bounds on regularity are better than Lvovsky's in certain cases.

Algorithms for checking Cohen--Macaulay and Buchsbaum properties are developed.

Application to sumsets structure analysis is demonstrated.

Abstract

A monomial curve $C$ is defined by a sequence of coprime integers $0 = a_0 < a_1 < \cdots < a_k =: d$. One gap of this sequence is $a_{i+1} - a_i - 1$. Gruson--Lazarsfeld--Peskine bound (1983) says that $reg (C) \le d - k +2$, which is equal to the sum of all gaps plus 2. Lvovsky (1996) showed that it is enough to take the sum of two largest gaps plus 2. In this paper, under some specific conditions, we give several new bounds which are better than Lvovsky's bound. Our method relies on the study of Apery sets and Frobenius numbers. From this we can give new criteria to check the (arithmetically) Cohen--Macaulay and Buchsbaum property of $C$. Algorithms are provided to check these properties as well as to compute $ reg(C)$ and other invariants. We also give an application to study the structure of sumsets.