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

Abstract

We study the number of lattice points in integer dilates of the rational polytope P ¼ ðx 1 ; . . . ; x n Þ 2 R n 50 :where a 1 ; . . . ; a n are positive integers. This polytope is closely related to the linear Diophantine problem of Frobenius: given relatively prime positive integers a 1 ; . . . ; a n ; find the largest value of t (the Frobenius number) such that m 1 a 1 þ Á Á Á þ m n a n ¼ t has no solution in positive integers m 1 ; . . . ; m n : This is equivalent to the problem of finding the largest dilate tP such that the facet f P n k¼1 x k a k ¼ tg contains no lattice point. We present two methods for computing the Ehrhart quasipolynomials Lð % P P; tÞ :¼ #ðtP \ Z n Þ and LðP8; tÞ :¼ #ðtP8 \ Z n Þ: Within the computations a Dedekind-like finite Fourier sum appears. We obtain a reciprocity law for these sums, generalizing a theorem of Gessel. As a corollary of our formulas, we rederive the reciprocity law for Zagier's higher-dimensional Dedekind sums. Finally, we find bounds for the Fourier-Dedekind sums and use them to give new bounds for the Frobenius number. # 2002 Elsevier Science (USA)

Integer Knapsacks: Average Behavior of the Frobenius Numbers

“…4) and by many other authors, as well as more recent results by Beck, Diaz, and Robins [6] g N ≤ 1 2 a 1 a 2 a 3 (a 1 + a 2 + a 3 ) − a 1 − a 2 − a 3 , (1.5)…”

Relations for Bernoulli–Barnes numbers and Barnes zeta functions

“…Our second main result expresses the Barnes zeta function in terms of Bernoulli-Barnes polynomials, Hurwitz zeta functions, and Fourier-Dedekind sums [5], defined as…”

An explicit formulation for two dimensional vector partition functions

“…When s = 1, an explicit formulation for t(b|M), which counts the integer solutions for the linear Diophantine equation, is presented in [1]. In particular,…”