ALGEBRAIC COMBINATORICS

no. 1
Commutative algebra of generalised Frobenius numbers
Manjunath, Madhusudan1; Smith, Ben2
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
Algebraic Combinatorics, Volume 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.

Received:
Accepted:
Published online:
DOI: 10.5802/alco.31
Classification: 11D07, 13D02, 52C07, 06A07
Keywords: 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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Manjunath, Madhusudan; Smith, Ben. Commutative algebra of generalised Frobenius numbers. Algebraic Combinatorics, Volume 2 (2019) no. 1, pp. 149-171. doi : 10.5802/alco.31. https://alco.centre-mersenne.org/articles/10.5802/alco.31/

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