New Identities for Degrees of Syzygies in Numerical Semigroups

TL;DR

This paper establishes new polynomial and quasipolynomial identities for the degrees of syzygies in the Hilbert series of numerical semigroups, enhancing understanding of their algebraic structure.

Contribution

It introduces novel identities for syzygy degrees in the Hilbert series of numerical semigroups, derived through rational and quasipolynomial representations.

Findings

Identities simplify for symmetric semigroups and complete intersections.

Provides a unified approach to studying syzygies in numerical semigroups.

Enhances algebraic understanding of semigroup Hilbert series.

Abstract

We derive a set of polynomial and quasipolynomial identities for degrees of syzygies in the Hilbert series H(d^m;z) of nonsymmetric numerical semigroups S(d^m) of arbitrary generating set of positive integers d^m={d_1,...,d_m}, m\geq 3. These identities were obtained by studying together the rational representation of the Hilbert series H(d^m;z) and the quasipolynomial representation of the Sylvester waves in the restricted partition function W(s,d^m). In the cases of symmetric semigroups and complete intersections these identities become more compact.