Solving thousand-digit Frobenius problems using Gröbner bases

Roune, Bjarke Hammersholt

doi:10.1016/j.jsc.2007.06.002

Cited by 30 publications

(10 citation statements)

References 17 publications

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“…It is fair to point out that Marcel Morales and Nguyen Thi Dung have recently produced an algorithm by using similar arguments for the computation of the Frobenius number (see [13]). Professor Morales has informed us that similar techniques were used by Einstein et al [8] and Roune [16] to give sophisticated algorithms for the computation of the Frobenius number of numerical semigroup. However, in these papers the important role of the Apery sets is not observed.…”

Section: Computation Of Apery Setsmentioning

confidence: 99%

The Short Resolution of a Semigroup Algebra

Ojeda¹,

Vigneron-Tenorio²

2017

Bull. Aust. Math. Soc.

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Abstract. This work generalizes the short resolution given in Proc. Amer. Math. Soc. 131, 4, (2003), 1081-1091, to any affine semigroup. Moreover, a characterization of Apéry sets is given. This characterization lets compute Apéry sets of affine semigroups and the Frobenius number of a numerical semigroup in a simple way. We also exhibit a new characterization of the Cohen-Macaulay property for simplicial affine semigroups.

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Section: Computation Of Apery Setsmentioning

confidence: 99%

The Short Resolution of a Semigroup Algebra

Ojeda¹,

Vigneron-Tenorio²

2017

Bull. Aust. Math. Soc.

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“…The computations of f (a) were performed using the Frobby software package by Roune[27]; cf. also[28]. We repeated the computations using other random seeds and/or changing T to 10 14 , as well as to 10 13 , 10 12 , 10 11 in some cases, and the resulting graphs were consistently found to be practically indistinguishable, except for d = 3 and R very near 2.…”

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confidence: 99%

On the limit distribution of Frobenius numbers

Strömbergsson

2012

Acta Arith.

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The Frobenius number g(a) of an integer vector a with positive coprime coefficients is defined as the largest integer that does not have a representation as a non-negative integer linear combination of the coefficients of a. According to a recent result by Marklof, if a is taken to be random in an expanding d-dimensional domain D, then (a1 · · · a d ) −1/(d−1) g(a) has a limit distribution. In the present paper we prove an asymptotic formula for the (algebraic) tail behavior of this limit distribution. We also prove that the corresponding upper bound on the probability of the Frobenius number being large holds uniformly with respect to the expansion factor of the domain D. Finally we prove that for large d, the limit distribution of (a1 · · · a d ) −1/(d−1) g(a) has almost all of its mass concentrated between (d − 1)! 1/(d−1) and 1.757 · (d − 1)! 1/(d−1) . The techniques involved in the proofs come from the geometry of numbers, and in particular we use results by Schmidt on the distribution of sublattices of Z m , and bounds by Rogers and Schmidt on lattice coverings of space with convex bodies.

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“…Also, Gröbner bases are being used to obtain efficient algorithms to solve the Frobenius problem. Following this via, we should cite the papers by Roune (see [25]), Márquez-Campos, Ojeda, and Tornero (see [15]), and the references therein.…”

Section: Introductionmentioning

confidence: 99%