Quasi-polynomial growth of numerical and affine semigroups with constrained gaps

TL;DR

This paper introduces a new geometric approach to counting numerical and affine semigroups, proving their counts grow as quasi-polynomials with respect to certain parameters, revealing underlying regularities.

Contribution

It develops a novel polyhedral framework using truncated addition tables to analyze semigroup enumeration, establishing quasi-polynomial growth results.

Findings

Number of numerical semigroups with fixed sporadic elements and Frobenius number is quasi-polynomial in the Frobenius number.

Generalization to affine semigroups shows similar quasi-polynomial growth in higher dimensions.

Application of Ehrhart's theorem to new polyhedral models provides a unified counting approach.

Abstract

A common tool in the theory of numerical semigroups is to interpret a desired class of semigroups as the integer lattice points in a rational polyhedron in order to leverage computational and enumerative techniques from polyhedral geometry. Most arguments of this type make use of a parametrization of numerical semigroups with fixed multiplicity $m$ in terms of their $m$-Ap\'{e}ry sets, giving a representation called Kunz coordinates which obey a collection of inequalities defining the Kunz polyhedron. In this work, we introduce a new class of polyhedra describing numerical semigroups in terms of a truncated addition table of their sporadic elements. Applying a classical theorem of Ehrhart to slices of these polyhedra, we prove that the number of numerical semigroups with $n$ sporadic elements and Frobenius number $f$ is polynomial up to periodicity, or quasi-polynomial, as a function of $f$ for fixed $n$. We also generalize this approach to higher dimensions to demonstrate quasi-polynomial growth of the number of affine semigroups with a fixed number of elements, and all gaps, contained in an integer dilation of a fixed polytope.