Genera of numerical semigroups and polynomial identities for degrees of syzygies

TL;DR

This paper develops polynomial identities for the degrees of syzygies in numerical semigroups, linking them to higher genera and Betti numbers, with implications for symmetry properties.

Contribution

It introduces a general framework of polynomial identities for syzygy degrees and relates the number of independent genera to semigroup symmetry and Betti numbers.

Findings

Identifies polynomial identities of arbitrary degree for syzygy degrees.

Shows that for n >= m, identities involve higher genera G_r.

Determines the number of algebraically independent genera g_m, depending on semigroup symmetry.

Abstract

We derive polynomial identities of arbitrary degree $n$ for syzygies degrees of numerical semigroups S_m=<d_1,...,d_m> and show that for n>=m they contain higher genera G_r=\sum_{s\in Z_>\setminus S_m}s^r of S_m. We find a number g_m=B_m-m+1 of algebraically independent genera G_r and equations, related any of g_m+1 genera, where B_m=\sum_{k=1}^{m-1}\beta_k and \beta_k denote the total and partial Betti numbers of non-symmetric semigroups. The number g_m is strongly dependent on symmetry of S_m and decreases for symmetric semigroups and complete intersections.