Commutative algebra of generalised Frobenius numbers

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.