Commutative Algebra of Generalised Frobenius Numbers

TL;DR

This paper explores the algebraic structure of generalized Frobenius numbers through lattice modules, providing explicit characterizations, finiteness results, and an algorithm for computation, thereby connecting number theory with commutative algebra.

Contribution

It introduces generalized lattice modules linked to Frobenius numbers, characterizes their generators, proves finiteness of their classes, and develops an algorithm for calculating these numbers.

Findings

The sequence of generalized Frobenius numbers forms a generalized arithmetic progression.

There are finitely many isomorphism classes of the associated lattice modules.

An explicit algorithm for computing the k-th Frobenius number is provided.

Abstract

We study commutative algebra arising from generalised Frobenius numbers. The $k$-th (generalised) Frobenius number of natural numbers $(a_1,\dots,a_n)$ is the largest natural number that cannot be written as a non-negative integral combination of $(a_1,\dots,a_n)$ in $k$ distinct ways. Suppose that $L$ is the lattice of integers points of $(a_1,\dots,a_n)^{\perp}$. Taking cue from the concept of lattice modules due to Bayer and Sturmfels, we define generalised lattice modules $M_L^{(k)}$ whose Castelnuovo-Mumford regularity captures the $k$-th Frobenius number of $(a_1,\dots,a_n)$. We study the sequence $\{M_L^{(k)}\}_{k=1}^{\infty}$ of generalised lattice modules providing an explicit characterisation of their minimal generators. We show that there are only finitely many isomorphism classes of generalized lattice modules. As a consequence of our commutative algebraic approach, we show that the sequence of generalised Frobenius numbers forms a generalised arithmetic progression. We also construct an algorithm to compute the $k$-th Frobenius number.