Intermediate Sums on Polyhedra: Computation and Real Ehrhart Theory

@article{Baldoni2010IntermediateSO,
  title={Intermediate Sums on Polyhedra: Computation and Real Ehrhart Theory},
  author={Velleda Baldoni and Nicole Berline and Matthias K{\"o}ppe and Mich{\`e}le Vergne},
  journal={ArXiv},
  year={2010},
  volume={abs/1011.6002}
}
We study intermediate sums, interpolating between integrals and discrete sums, which were introduced by A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), 1449--1466]. For a given semi-rational polytope P and a rational subspace L, we integrate a given polynomial function h over all lattice slices of the polytope P parallel to the subspace L and sum up the integrals. We first develop an algorithmic theory of parametric intermediate generating… 

Figures and Tables from this paper

Computation of the Highest Coefficients of Weighted Ehrhart Quasi-polynomials of Rational Polyhedra
TLDR
An efficient algorithm for computing the highest degree coefficients of the weighted Ehrhart quasi-polynomial for a rational simple polytope in varying dimension, when the weights of the lattice points are given by a polynomial function h.
INTERMEDIATE SUMS ON POLYHEDRA II: BIDEGREE AND POISSON FORMULA
We continue our study of intermediate sums over polyhedra, interpolating between integrals and discrete sums, which were introduced by Barvinok [Computing the Ehrhart quasi-polynomial of a rational
Boundary $h^\ast$-polynomials of rational polytopes
If P is a lattice polytope (i.e., P is the convex hull of finitely many integer points in R), Ehrhart’s famous theorem asserts that the integer-point counting function |nP X Z| is a polynomial in the
Rational Ehrhart Theory
The Ehrhart quasipolynomial of a rational polytope P encodes fundamental arithmetic data of P, namely, the number of integer lattice points in positive integral dilates of P. Ehrhart quasipolynomials
ON THE COEFFICIENTS OF RATIONAL EHRHART QUASI-POLYNOMIALS OF
By extending former results of Ehrhart, it was shown by Peter McMullen that the number of lattice points in the Minkowski-sum of dilated rational polytopes is a quasipolynomial function in the
Lattice Points in Algebraic Cross-polytopes and Simplices
  • B. Borda
  • Mathematics
    Discret. Comput. Geom.
  • 2018
TLDR
The main ingredients of the proof are a Poisson summation formula for general algebraic polytopes, and a representation of the Fourier transform of the characteristic function of an arbitrary simplex in the form of a complex line integral.
Ehrhart theory for real dilates of polytopes
ROYER, T. Ehrhart theory for real dilates of polytopes. 2017. 80 pp. Master Thesis — Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, 2017. The Ehrhart function LP (t) of
Dedekind Sums, the Building Blocks of Lattice-Point Enumeration
We encountered Dedekind sums in our study of finite Fourier analysis in Chapter 7, and we became intimately acquainted with their siblings in our study of the coin-exchange problem in Chapter 1 They
7: Lattice points and lattice polytopes
Lattice polytopes arise naturally in algebraic geometry, analysis, combinatorics, computer science, number theory, optimization, probability and representation theory. They possess a rich structure
Rational Ehrhart quasi-polynomials
  • Eva Linke
  • Mathematics
    J. Comb. Theory, Ser. A
  • 2011
...
...

References

SHOWING 1-10 OF 21 REFERENCES
Computation of the Highest Coefficients of Weighted Ehrhart Quasi-polynomials of Rational Polyhedra
TLDR
An efficient algorithm for computing the highest degree coefficients of the weighted Ehrhart quasi-polynomial for a rational simple polytope in varying dimension, when the weights of the lattice points are given by a polynomial function h.
Counting with rational generating functions
Computing the Ehrhart quasi-polynomial of a rational simplex
TLDR
A polynomial time algorithm to compute any fixed number of the highest coefficients of the Ehrhart quasi-polynomial of a rational simplex for rational polytopes of a fixed dimension is presented.
Residue formulae, vector partition functions and lattice points in rational polytopes
We obtain residue formulae for certain functions of several vari- ables. As an application, we obtain closed formulae for vector partition func- tions and for their continuous analogs. They imply an
An Algorithmic Theory of Lattice Points in Polyhedra
We discuss topics related to lattice points in rational polyhedra, including efficient enumeration of lattice points, “short” generating functions for lattice points in rational polyhedra, relations
Computing Parametric Rational Generating Functions with a Primal Barvinok Algorithm
TLDR
It is proved that, on the level of indicator functions of polyhedra, there is no need for using inclusion–exclusion formulas to account for boundary effects, and all linear identities in the space of indicator function identities can be purely expressed using partially open variants of the full-dimensionalpolyhedra in the identity.
How to integrate a polynomial over a simplex
TLDR
It is proved that the problem is NP-hard for arbitrary polynomials via a generalization of a theorem of Motzkin and Straus, and if the polynomial depends only on a fixed number of variables, while its degree and the dimension of the simplex are allowed to vary, it is proven that integration can be done inPolynomial time.
Rational Ehrhart quasi-polynomials
  • Eva Linke
  • Mathematics
    J. Comb. Theory, Ser. A
  • 2011
Local Euler-Maclaurin expansion of Barvinok valuations and Ehrhart coefficients of a rational polytope
We extend to Barvinok's valuations the Euler-Maclaurin expansion formula which we obtained previously for the sum of values of a polynomial over the integral points of a rational polytope. This leads
Reverse Search for Enumeration
...
...