Frobenius number and minimum genus of numerical semigroups with fixed multiplicity and embedding dimension

TL;DR

This paper introduces algorithms to compute the minimal Frobenius number and genus for numerical semigroups with fixed multiplicity and embedding dimension, also identifying semigroups where these minima occur.

Contribution

It provides new algorithms for calculating minimal Frobenius number and genus in numerical semigroups with specified parameters, and identifies semigroups achieving these minima.

Findings

Algorithms for minimal Frobenius number and genus are developed.

Semigroups achieving minimal values are explicitly characterized.

The methods facilitate analysis of numerical semigroups with fixed parameters.

Abstract

Fixed two positive integers m and e, some algorithms for computing the minimal Frobenius number and minimal genus of the set of numerical semigroups with multiplicity m and embedding dimension e are provided. Besides, the semigroups where these minimal values are achieved are computed too.