Commutative algebra of generalised Frobenius numbers

Manjunath, Madhusudan; Smith, Ben

Manjunath, Madhusudan ; Smith, Ben

Algebraic Combinatorics, Volume 2 (2019) no. 1, p. 149-171

Abstract

We study commutative algebra arising from generalised Frobenius numbers. The

k

-th (generalised) Frobenius number of relatively prime natural numbers

(a_{1}, \dots, a_{n})

is the largest natural number that cannot be written as a non-negative integral combination of

(a_{1}, \dots, a_{n})

k

distinct ways. Suppose that

L

is the lattice of integer points of

{(a_{1}, \dots, 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}, \dots, a_{n})

. We study the sequence

{M_{L}^{(k)}}_{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.

Received : 2018-07-03
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},
     publisher = {MathOA foundation},
     volume = {2},
     number = {1},
     year = {2019},
     pages = {149-171},
     doi = {10.5802/alco.31},
     language = {en},
     url = {https://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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