The Frobenius Problem, Rational Polytopes, and Fourier–Dedekind Sums,

Beck, Matthias; Diaz, Ricardo; Robins, Sinai

doi:10.1016/s0022-314x(02)92786-1

Cited by 14 publications

(22 citation statements)

References 19 publications

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“…No explicit formula exists when S consists of at least three distinct numbers. On the positive side, upper bounds exist (see [BDR02] and [FR07] for some estimates and further references) as do polynomial time algorithms determining the Frobenius number for sets S of fixed size [Kan92].…”

Section: Definition 1 (Frobenius Values)mentioning

confidence: 99%

Frobenius Problem and dead ends in integers

Sunic¹

2008

Journal of Number Theory

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Let a and b be positive and relatively prime integers. We show that the following are equivalent: (i) d is a dead end in the (symmetric) Cayley graph of Z with respect to a and b, (ii) d is a Frobenius value with respect to a and b (it cannot be written as a non-negative or non-positive integer linear combination of a and b), and d is maximal (in the Cayley graph) with respect to this property. In addition, for given integers a and b, we explicitly describe all such elements in Z. Finally, we show that Z has only finitely many dead ends with respect to any finite symmetric generating set. In Appendix A we show that every finitely generated group has a generating set with respect to which dead ends exist.

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Section: Definition 1 (Frobenius Values)mentioning

confidence: 99%

Frobenius Problem and dead ends in integers

Sunic¹

2008

Journal of Number Theory

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“…Among many others, they include results by Beck et al [4] produced with the use of bounds on Fourier-Dedekind sums:…”

Section: Fukshansky and S Robinsmentioning

confidence: 99%

“…See [4] for further bibliography. For comparison, here is a lower bound on F by Aliev and Gruber [1]:…”

Section: Fukshansky and S Robinsmentioning

confidence: 99%

Frobenius Problem and the Covering Radius of a Lattice

Fukshansky

Robins

2007

Discrete Comput Geom

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Let N ≥ 2 and let 1 < a 1 < · · · < a N be relatively prime integers. The Frobenius number of this N -tuple is defined to be the largest positive integer that cannot be expressed as N i=1 a i x i where x 1 , . . . , x N are non-negative integers. The condition that gcd(a 1 , . . . , a N ) = 1 implies that such a number exists. The general problem of determining the Frobenius number given N and a 1 , . . . , a N is NP-hard, but there have been a number of different bounds on the Frobenius number produced by various authors. We use techniques from the geometry of numbers to produce a new bound, relating the Frobenius number to the covering radius of the null-lattice of this N -tuple. Our bound is particularly interesting in the case when this lattice has equal successive minima, which, as we prove, happens infinitely often.

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“…The most recent result is from BeckDiaz-Robins [4,6]. Applying the residue theorem to the case of rational right-angled simplices, they found formulas involving generalized Dedekind sums.…”

Section: Introductionmentioning

confidence: 99%

On the GLY Conjecture of upper estimate of positive integral points in real right-angled simplices

Wang

Yau

2007

Journal of Number Theory

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The GLY (Granville-Lin-Yau) Conjecture is a generalization of Lin, Xu and Yau's results. An important application of GLY is its use in characterizing an affine hypersurface in C n as a cone over a nonsingular projective variety. In addition, the Rough Upper Estimate Conjecture in GLY, recently proved by Yau and Zhang, implies the Durfee Conjecture in singularity theory. This paper develops a unified approach to prove the Sharp Upper Estimate Conjecture for general n. Using this unified approach, we prove that the Sharp Upper Estimate Conjecture is true for n = 4, 5, 6. After giving a counter-example to show that the Sharp Upper Estimate Conjecture is not true for n = 7, we propose a Modified GLY Conjecture. For each fixed n, our unified approach can be used to prove this Modified GLY Conjecture.

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The Frobenius Problem, Rational Polytopes, and Fourier–Dedekind Sums,

Cited by 14 publications

References 19 publications

Frobenius Problem and dead ends in integers

Frobenius Problem and dead ends in integers

Frobenius Problem and the Covering Radius of a Lattice

On the GLY Conjecture of upper estimate of positive integral points in real right-angled simplices

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