Foundations of Computational Mathematics

, Volume 12, Issue 4, pp 435–469 | Cite as

Computation of the Highest Coefficients of Weighted Ehrhart Quasi-polynomials of Rational Polyhedra

  • V. Baldoni
  • N. Berline
  • J. A. De Loera
  • M. Köppe
  • M. Vergne


This article concerns the computational problem of counting the lattice points inside convex polytopes, when each point must be counted with a weight associated to it. We describe 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. Our technique is based on a refinement of an algorithm of A. Barvinok in the unweighted case (i.e., h≡1). In contrast to Barvinok’s method, our method is local, obtains an approximation on the level of generating functions, handles the general weighted case, and provides the coefficients in closed form as step polynomials of the dilation. To demonstrate the practicality of our approach, we report on computational experiments which show that even our simple implementation can compete with state-of-the-art software.


Ehrhart functions Exponential sums and integrals Intermediate sums Polynomial-time algorithms 

Mathematics Subject Classification (2010)

05A15 52C07 68R05 68U05 52B20 


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Copyright information

© SFoCM 2011

Authors and Affiliations

  • V. Baldoni
    • 1
  • N. Berline
    • 2
  • J. A. De Loera
    • 3
  • M. Köppe
    • 3
  • M. Vergne
    • 4
  1. 1.Dipartimento di MatematicaUniversità degli studi di Roma “Tor Vergata”RomaItaly
  2. 2.Centre de Mathématiques Laurent SchwartzÉcole PolytechniquePalaiseau CedexFrance
  3. 3.Department of MathematicsUniversity of California, DavisDavisUSA
  4. 4.Théorie des GroupesInstitut de Mathématiques de JussieuParis Cedex 05France

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