The Maximal Denumerant of a Numerical Semigroup

TL;DR

This paper introduces an algorithm to compute the maximal denumerant of a numerical semigroup, linking it to algebraic properties like Cohen-Macaulay and Gorenstein conditions of associated graded rings.

Contribution

It provides a novel algorithm for calculating the maximal denumerant and explores its connections to important algebraic properties of related rings.

Findings

Algorithm efficiently computes the maximal denumerant.

Connections established between maximal denumerant and Cohen-Macaulay properties.

Simplifications of the algorithm in special algebraic cases.

Abstract

Given a numerical semigroup S = <a_0, a_1, a_2,..., a_t> and n in S, we consider the factorization n = c_0 a_0 + c_1 a_1 + ... + c_t a_t where c_i >= 0. Such a factorization is maximal if c_0 + c_1 + ... + c_t is a maximum over all such factorizations of n. We provide an algorithm for computing the maximum number of maximal factorizations possible for an element in S, which is called the maximal denumerant of S. We also consider various cases that have connections to the Cohen-Macualay and Gorenstein properties of associated graded rings for which this algorithm simplifies.