The Expected Embedding Dimension, type and weight of a Numerical Semigroup

TL;DR

This paper investigates the asymptotic properties of numerical semigroups, showing that as the genus increases, most have embedding dimension, type, and weight close to specific proportional values, revealing their typical structure.

Contribution

It establishes that for large genus, numerical semigroups predominantly have embedding dimension, type, and weight near certain proportional constants, answering a question posed by Eliahou.

Findings

Proportion of semigroups with embedding dimension close to g/√5 approaches 1 as g→∞

Similar asymptotic results are proven for type and weight of semigroups

Provides insight into the typical structure of large genus numerical semigroups

Abstract

We study statistical properties of numerical semigroups of genus $g$ as $g$ goes to infinity. More specifically, we answer a question of Eliahou by showing that as $g$ goes to infinity, the proportion of numerical semigroups of genus $g$ with embedding dimension close to $g/\sqrt{5}$ approaches $1$. We prove similar results for the type and weight of a numerical semigroup of genus $g$.