A note on the computation of the Frobenius number of a numerical semigroup

TL;DR

This paper presents a novel algebraic approach to computing the Frobenius number of a numerical semigroup using the maximal socle degree of a specific quotient algebra, offering a potentially more efficient method.

Contribution

It introduces a new algebraic characterization of the Frobenius number via socle degrees, connecting semigroup theory with algebraic properties of quotient algebras.

Findings

Frobenius number can be derived from socle degree of a quotient algebra.

Provides an algebraic method for computing the conductor of a numerical semigroup.

Establishes a link between semigroup invariants and algebraic structures.

Abstract

In this note we observe that the Frobenius number and therefore the conductor of a numerical semigroup can be obtained from the maximal socle degree of the quotient of the corresponding semigroup algebra by the ideal generated by the biggest generator of the semigroup.