Commutative algebra of generalised Frobenius numbers

Ben Smith, Madhusudan Manjunath

Research output: Contribution to journalArticlepeer-review


We study commutative algebra arising from generalised Frobenius numbers. The k-th (generalised) Frobenius number of relatively prime natural numbers (a1,⋯,an) is the largest natural number that cannot be written as a non-negative integral combination of (a1,⋯,an) in k distinct ways. Suppose that L is the lattice of integer points of (a1,⋯,an)⊥. Taking our cue from the concept of lattice modules due to Bayer and Sturmfels, we define generalised lattice modules M(k)L whose Castelnuovo–Mumford regularity captures the k-th Frobenius number of (a1,⋯,an). We study the sequence {M(k)L}∞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.
Original languageEnglish
Pages (from-to)149-171
JournalAlgebraic Combinatorics
Issue number1
Publication statusPublished - 4 Feb 2019


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