We study commutative algebra arising from generalised Frobenius numbers. The -th (generalised) Frobenius number of relatively prime natural numbers is the largest natural number that cannot be written as a non-negative integral combination of in distinct ways. Suppose that is the lattice of integer points of . Taking our cue from the concept of lattice modules due to Bayer and Sturmfels, we define generalised lattice modules whose Castelnuovo–Mumford regularity captures the -th Frobenius number of . We study the sequence 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 -th Frobenius number.
Accepted: 2018-07-04
Published online: 2019-02-04
DOI: https://doi.org/10.5802/alco.31
Classification: 11D07, 13D02, 52C07, 06A07
Keywords: Frobenius number, syzygy, lattice, poset, Castelnuovo–Mumford regularity
@article{ALCO_2019__2_1_149_0, author = {Manjunath, Madhusudan and Smith, Ben}, title = {Commutative algebra of generalised Frobenius numbers}, journal = {Algebraic Combinatorics}, pages = {149--171}, publisher = {MathOA foundation}, volume = {2}, number = {1}, year = {2019}, doi = {10.5802/alco.31}, mrnumber = {3912171}, zbl = {07024222}, language = {en}, url = {alco.centre-mersenne.org/item/ALCO_2019__2_1_149_0/} }
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/item/ALCO_2019__2_1_149_0/
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