The Frobenius Problem, Rational Polytopes, and Fourier-Dedekind Sums1 (original) (raw)
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Discrete & Computational Geometry, 2002
We give explicit, polynomial-time computable formulas for the number of integer points in any two-dimensional rational polygon. A rational polygon is one whose vertices have rational coordinates. We find that the basic building blocks of our formulas are Dedekind-Rademacher sums, which are polynomial-time computable finite Fourier series. As a by-product we rederive a reciprocity law for these sums due to Gessel, which generalizes the reciprocity law for the classical Dedekind sums. In addition, our approach shows that Gessel's reciprocity law is a special case of the one for Dedekind-Rademacher sums, due to Rademacher. The full beauty of the subject of generating functions emerges only from tuning in on both channels: the discrete and the continuous. Herb Wilf [W, p. vii] * Parts of this work appeared in the first author's Ph.D. thesis. The second author kindly acknowledges the support of NSA Grant MSPR-OOY-196.
Refined upper bounds for the linear Diophantine problem of Frobenius
Advances in Applied Mathematics, 2004
We study the Frobenius problem: given relatively prime positive integers a1,. .. , a d , find the largest value of t (the Frobenius number g(a1,. .. , a d)) such that d k=1 m k a k = t has no solution in nonnegative integers m1,. .. , m d. We introduce a method to compute upper bounds for g(a1, a2, a3), which seem to grow considerably slower than previously known bounds. Our computations are based on a formula for the restricted partition function, which involves Dedekind-Rademacher sums, and the reciprocity law for these sums.
FROBENIUS NUMBERS BY LATTICE POINT ENUMERATION
The Frobenius number g(A) of a set A = (a 1 , a 2 , . . . , a n ) of positive integers is the largest integer not representable as a nonnegative linear combination of the a i . We interpret the Frobenius number in terms of a discrete tiling of the integer lattice of dimension n−1 and obtain a fast algorithm for computing it. The algorithm appears to run in average time that is softly quadratic and we prove that this is the case for almost all of the steps. In practice, the algorithm is very fast: examples with n = 4 and the numbers in A having 100 digits take under one second. The running time increases with dimension and we can succeed up to n = 11.
On lattice point counting in Δ-modular polyhedra
arXiv (Cornell University), 2020
Let a polyhedron P be defined by one of the following ways: (i) P = {x ∈ R n : Ax ≤ b}, where A ∈ Z (n+k)×n , b ∈ Z (n+k) and rank A = n, (ii) P = {x ∈ R n + : Ax = b}, where A ∈ Z k×n , b ∈ Z k and rank A = k, and let all rank order minors of A be bounded by ∆ in absolute values. We show that the short rational generating function for the power series m∈P ∩Z n x m can be computed with the arithmetical complexity O T SNF (d) • d k • d log 2 ∆ , where k and ∆ are fixed, d = dim P , and T SNF (m) is the complexity of computing the Smith Normal Form for m × m integer matrices. In particular, d = n, for the case (i), and d = n − k, for the case (ii). The simplest examples of polyhedra that meet the conditions (i) or (ii) are the simplices, the subset sum polytope and the knapsack or multidimensional knapsack polytopes. Previously, the existence of a polynomial time algorithm in varying dimension for the considered class of problems was unknown already for simplicies (k = 1). We apply these results to parametric polytopes and show that the step polynomial representation of the function c P (y) = |P y ∩ Z n |, where P y is a parametric polytope, whose structure is close to the cases (i) or (ii), can be computed in polynomial time even if the dimension of P y is not fixed. As
Weighted Lattice Point Sums in Lattice Polytopes, Unifying Dehn–Sommerville and Ehrhart–Macdonald
Discrete & Computational Geometry
Let V be a real vector space of dimension n and let M ⊂ V be a lattice. Let P ⊂ V be an n-dimensional polytope with vertices in M , and let ϕ : V → C be a homogeneous polynomial function of degree d. For q ∈ Z >0 and any face F of P , let D ϕ,F (q) be the sum of ϕ over the lattice points in the dilate qF. We define a generating function G ϕ (q, y) ∈ Q[q][y] packaging together the various D ϕ,F (q), and show that it satisfies a functional equation that simultaneously generalizes Ehrhart-Macdonald reciprocity and the Dehn-Sommerville relations. When P is a simple lattice polytope (i.e., each vertex meets n edges), we show how G ϕ can be computed using an analogue of Brion-Vergne's Euler-Maclaurin summation formula. (For instance, if P is a simplex, then h(P, t) = t n + t n−1 + • • • + 1.) The Dehn-Sommerville relations say that h k (P) = h n−k (P) for all k.
An analogue of the Rademacher function for generalized Dedekind sums in higher dimension
We consider generalized Dedekind sums in dimension n, for fixed n-tuple of natural numbers, defined as sum of products of values of periodic Bernoulli functions. This includes the higher dimensional Dedekind sums of Zagier and Apostol-Carlitz' generalized Dedekind sums as well as the original Dedekind sums. These are realized as coefficients of Todd series of lattice cones and satisfy reciprocity law from the cocycle property of Todd series. Using iterated residue formula, we compute the coefficient of the decomposition of of the Todd series corresponding to a nonsingular decomposition of the lattice cone defining the Dedekind sums. We associate a Laurent polynomial which is added to generalized Dedekind sums of fixed index to make their denominators bounded. We give explicitly the denominator in terms of Bernoulli numbers. This generalizes the role played by the rational function given by the difference of the Rademacher function and the classical Dedekind sums. We associate an exponential sum to the generalized Dedekind sums using the integrality of the generalized Rademacher function. We show that this exponential sum has a nontrivial bound that is sufficient to fulfill Weyl's equidistribution criterion and thus the fractional part of the generalized Dedekind sums are equidistributed. As an example, for a 3 dimensional case and Zagier's higher dimensional generalization of Dedekind sums, we compute the Laurent polynomials associated. CONTENTS HI-JOON CHAE, BYUNGHEUP JUN, AND JUNGYUN LEE 1.6. Main result 9 Notations 12 2. Todd series 13 2.1. Lattice cones 13 2.2. Chain complex of lattice cones 14 2.3. Dual cones 15 2.4. Todd series 15 2.5. Todd cocycle 16 3. Dedekind sums and Todd coeffients 17 4. Integrality of generalized Dedekind sums 21 5. Reduction mod q of generalized Dedekind sums 25 6. Equidistribution of generalized Dedekind sums and exponential sums 31 7. Examples 36 7.1. Three dimensional Dedekind sums 36 7.2. Dedekind-Zagier sums 36 Appendix A. Number of congruence solutions modulo a prime power 37 Appendix B. Iterated constant term and multidimensional residue in general coefficient 38 References 42 DEPARTMENT OF MATHEMATICS EDUCATION, HONGIK UNIVERSITY, SEOUL 121-791, REPUBLIC OF KOREA
Effective lattice point counting in rational convex polytopes
Journal of Symbolic Computation, 2004
This paper discusses algorithms and software for the enumeration of all lattice points inside a rational convex polytope: we describe LattE, a computer package for lattice point enumeration which contains the first implementation of A. Barvinok's algorithm (Math. Oper. Res. 19 (1994)