Residue formulae, vector partition functions and lattice points in rational polytopes

We obtain residue formulae for certain functions of several variables. As an application, we obtain closed formulae for vector partition functions and for their continuous analogs. They imply an Euler-MacLaurin summation formula for vector partition functions, and for rational convex polytopes as well: we express the sum of values of a polynomial function at all lattice points of a rational convex polytope in terms of the variation of the integral of the function over the deformed polytope.

     

Primitive lattice points inside an ellipse

Let Q(u, v) be a positive definite binary quadratic form with arbitrary real coefficients. For large real x, one may ask for the number B(x) of primitive lattice points (integer points (m, n) with gcd(M, n) = 1) in the ellipse disc Q(u, v) x, in particular, for the remainder term R(x) in the asymptotics for B(x). While upper bounds for R(x) depend on zero-free regions of the zeta-function, and thus, in most published results, on the Riemann Hypothesis, the present paper deals with a lower estimate. It is proved that the absolute value or R(x) is, in integral mean, at least a positive constant c time x ^1/4. Furthermore, it is shown how to find an explicit value for c, for each specific given form Q.To Professor Ekkehard Kratzel on his 70th birthday     

Splines,lattice points,and arithmetic matroids

Let X be a \((d\times N)\)-matrix. We consider the variable polytope \(\varPi _X(u) = \{ w \ge 0 : X w = u \}\). It is known that the function \(T_X\) that assigns to a parameter \(u \in \mathbb {R}^d\) the volume of the polytope \(\varPi _X(u)\) is piecewise polynomial. The Brion–Vergne formula implies that the number of lattice points in \(\varPi _X(u)\) can be obtained by applying a certain differential operator to the function \(T_X\). In this article, we slightly improve the Brion–Vergne formula and we study two spaces of differential operators that arise in this context: the space of relevant differential operators (i.e. operators that do not annihilate \(T_X\)) and the space of nice differential operators (i.e. operators that leave \(T_X\) continuous). These two spaces are finite-dimensional homogeneous vector spaces, and their Hilbert series are evaluations of the Tutte polynomial of the arithmetic matroid defined by the matrix X. They are closely related to the \(\mathscr {P}\)-spaces studied by Ardila–Postnikov and Holtz–Ron in the context of zonotopal algebra and power ideals.     

Short rational generating functions for lattice point problems

We prove that for any fixed the generating function of the projection of the set of integer points in a rational -dimensional polytope can be computed in polynomial time. As a corollary, we deduce that various interesting sets of lattice points, notably integer semigroups and (minimal) Hilbert bases of rational cones, have short rational generating functions provided certain parameters (the dimension and the number of generators) are fixed. It follows then that many computational problems for such sets (for example, finding the number of positive integers not representable as a non-negative integer combination of given coprime positive integers ) admit polynomial time algorithms. We also discuss a related problem of computing the Hilbert series of a ring generated by monomials.

     

Holomorphic Hermite functions and ellipses

In 1990 van Eijndhoven and Meyers introduced systems of holomorphic Hermite functions and reproducing kernel Hilbert spaces associated with the systems on the complex plane. Moreover they studied the relationship between the family of all their Hilbert spaces and a class of Gelfand–Shilov functions. After that, their systems of holomorphic Hermite functions have been applied to studying quantization on the complex plane, combinatorics, and etc. On the other hand, the author recently introduced systems of holomorphic Hermite functions associated with ellipses on the complex plane. The present paper shows that their systems of holomorphic Hermite functions are determined by some cases of ellipses, and that their reproducing kernel Hilbert spaces are some cases of the Segal–Bargmann spaces determined by the Bargmann-type transforms introduced by Sjöstrand.     

Repelling periodic points of given periods of rational functions

Let R(z) be a rational function of degree d≥2. Then R(z) has at least one repelling periodic point of given period k≥2, unless k = 4 and d = 2, or k = 3 and d≤3, or k = 2 and d≤8. Examples show that all exceptional cases occur.     

Decomposition and interval arithmetic applied to global minimization of polynomial and rational functions

A recent global optimization algorithm using decomposition (GOP), due to Floudas and Visweswaran, when specialized to the case of polynomial functions is shown to be equivalent to an interval arithmetic global optimization algorithm which applies natural extension to the cord-slope form of Taylor's expansion. Several more efficient variants using other forms of interval arithmetic are explored. Extensions to rational functions are presented. Comparative computational experiences are reported.     

Ladder operators for Szego polynomials and related biorthogonal rational functions

We find the raising and lowering operators for orthogonal polynomials on the unit circle introduced by Szego and for their four parameter generalization to biorthogonal rational functions on the unit circle.

     

On the exact location of the zeros of certain families of rational period functions and other related rational functions

The classification of Rational Period Functions on the modular group has been of some interest recently, and was accomplished by studying the pole sets of these rational functions. We take a complex analytic point of view and begin an investigation into the location of zeros of certain families of rational period functions.

     

Gap orders of rational functions on plane curves with few singular points

We prove that certain integers n cannot occur as degrees of linear series without base points on the normalization of a plane curve whose only singularities are a “small” number of nodes and ordinary cusps. As a consequence we compute the gonality of such a curve. Work done with financial support of M.U.R.S.T. while the authors were members of C.N.R.     

Lattice points in rational ellipsoids

We combine exponential sums, character sums and Fourier coefficients of automorphic forms to improve the best known upper bound for the lattice error term associated to rational ellipsoids.     

Primitive iteration and unary functions

Programming practice suggests a general notion of primitive iteration which subsumes the for-until-do construct as well as all known primative iteration operators. This leads to new iterative characterizations of primitive computable functions usable in computer science.     

Lattice points in lattice polytopes

IfK is the underlying point-set of a simplicial complex of dimension at mostd whose vertices are lattice points, and ifG(K) is the number of lattice points inK, then the lattice point enumeratorG(K,t)=1+ _n1 G(nK)t ⁿ takes the formC(K, t)/(1–t) ^d+1, for some polynomialC(K, t). Here,C(K, t) is expressed as a sum of local terms, one for each face ofK. WhenK is a polytope or its boundary, there result inequalities between the numbersG _r (K), whereG(n K)= _r=0 ^d n ^r G _r (K).