Duality Relation for the Hilbert Series of Almost Symmetric Numerical Semigroups

TL;DR

This paper establishes a duality relation for the Hilbert series of almost symmetric numerical semigroups, linking syzygy degrees, gaps, and generators, and applies it to characterize semigroups of maximal embedding dimension.

Contribution

It introduces a duality relation for the Hilbert series of almost symmetric semigroups and provides conditions for generating semigroups of maximal embedding dimension.

Findings

Derived the duality relation for the Hilbert series.

Established a bijection between syzygy degrees and gaps.

Provided conditions for generating semigroups of maximal embedding dimension.

Abstract

We derive the duality relation for the Hilbert series H(d^m;z) of almost symmetric numerical semigroup S(d^m) combining it with its dual H(d^m;z^{-1}). On this basis we establish the bijection between the multiset of degrees of the syzygy terms and the multiset of the gaps F_j, generators d_i and their linear combinations. We present the relations for the sums of the Betti numbers of even and odd indices separately. We apply the duality relation to the simple case of the almost symmetric semigroups of maximal embedding dimension, and give the necessary and efficient conditions for minimal set d^m to generate such semigroups.