We study commutative algebra arising from generalised Frobenius numbers. The $k$-th (generalised) Frobenius number of relatively prime natural numbers $({a}_{1},\cdots ,{a}_{n})$ is the largest natural number that cannot be written as a non-negative integral combination of $({a}_{1},\cdots ,{a}_{n})$ in $k$ distinct ways. Suppose that $L$ is the lattice of integer points of ${({a}_{1},\cdots ,{a}_{n})}^{\perp}$. Taking our cue from the concept of lattice modules due to Bayer and Sturmfels, we define generalised lattice modules ${M}_{L}^{\left(k\right)}$ whose Castelnuovo–Mumford regularity captures the $k$-th Frobenius number of $({a}_{1},\cdots ,{a}_{n})$. We study the sequence ${\left\{{M}_{L}^{\left(k\right)}\right\}}_{k=1}^{\infty}$ 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.

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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