Commutative algebra of generalised Frobenius numbers
Algebraic Combinatorics, Tome 2 (2019) no. 1, pp. 149-171.

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.

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DOI : 10.5802/alco.31
Classification : 11D07, 13D02, 52C07, 06A07
Mots clés : Frobenius number, syzygy, lattice, poset, Castelnuovo–Mumford regularity
Manjunath, Madhusudan 1 ; Smith, Ben 2

1 Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
2 School of Mathematical Sciences, Queen Mary University of London, London, E1 4NS, United Kingdom
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Manjunath, Madhusudan; Smith, Ben. Commutative algebra of generalised Frobenius numbers. Algebraic Combinatorics, Tome 2 (2019) no. 1, pp. 149-171. doi : 10.5802/alco.31. http://www.numdam.org/articles/10.5802/alco.31/

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