Commutative algebra and the linear diophantine problem of Frobenius

TL;DR

This paper explores the Frobenius problem using commutative algebra, providing algebraic proofs for classical results on Frobenius numbers and genus for two-element sets.

Contribution

It introduces a novel algebraic approach to the Frobenius problem, connecting semigroup theory with graded rings and Hilbert's syzygy theorem.

Findings

Algebraic proof of Frobenius number for two elements

Connection between semigroup properties and graded rings

New insights into the structure of numerical semigroups

Abstract

Let $A$ be a finite set of relatively prime positive integers, and let $S(A)$ be the set of all nonnegative integral linear combinations of elements of $A$. The set $S(A)$ is a semigroup that contains all sufficiently large integers. The largest integer not in $S(A)$ is the Frobenius number of $A$, and the number of positive integers not in $S(A)$ is the genus of $A$. Sharp and Sylvester proved in 1884 that the Frobenius number of the set $A = \{a,b\}$ is $ab-a-b$, and that the genus of $A$ is $(a-1)(b-1)/2$. Graded rings and a simple form of Hilbert's syzygy theorem are used to give a commutative algebra proof of this result.