Commutative algebra of generalised Frobenius numbers

Madhusudan Manjunath,Ben Smith

doi:10.5802/alco.31

Abstract

We study commutative algebra arising from generalised Frobenius numbers. The k-th (generalised) Frobenius number of relatively prime natural numbers (a 1 ,⋯,a n ) is the largest natural number that cannot be written as a non-negative integral combination of (a 1 ,⋯,a n ) in k distinct ways. Suppose that L is the lattice of integer points of (a 1 ,⋯,a n ) ⊥ . Taking our 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 ,⋯,a n ). We study the sequence {M L (k) } k=1 ∞ of generalised lattice modules providing an explicit characterisation of their minimal generators. We show that there are only finitely many isomorphism classes of generalised lattice modules. As a consequence of our commutative algebraic approach, we show that the sequence of generalised Frobenius numbers forms a finite difference progression i.e. a sequence whose set of successive differences is finite. We also construct an algorithm to compute the k-th Frobenius number.

Highlights

The Frobenius number F (a1, . . . , an) of a collection (a1, . . . , an) of natural numbers such that gcd(a1, . . . , an) = 1 is the largest natural number that cannot be expressed as a non-negative integral linear combination of a1, . . . , an
The Frobenius number is precisely the largest integer r such that there exists a point p ∈ (Zn) that evaluates to r at (a1, . . . , an) and p does not dominate any point in L(a1, . . . , an)
As an application of Theorem 1.12, we show that the sequence of Frobenius numbers (Fk)∞ k=1 is a finite difference progression

Summary

Introduction

An) of natural numbers such that gcd(a1, . An) = 1 is the largest natural number that cannot be expressed as a non-negative integral linear combination of a1, . The Frobenius number can be rephrased in the language of lattices as follows [18]. An) ∈ Zn. The Frobenius number is precisely the largest integer r such that there exists a point p ∈ (Zn) that evaluates to r at The domination is according to the partial order induced by the standard basis on (Zn). This leads to a commutative algebraic interpretation of the Frobenius number that we recall.

Generalised Lattice Modules

Finiteness Results

Applications

Future Directions