Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-26T08:32:07.262Z Has data issue: false hasContentIssue false
  • English
  • Français

INTERMEDIATE SUMS ON POLYHEDRA: COMPUTATION AND REAL EHRHART THEORY

Part of: Discrete mathematics in relation to computer science Polytopes and polyhedra Discrete geometry Combinatorics

Published online by Cambridge University Press:  05 September 2012

V. Baldoni ,
N. Berline ,
M. Köppe  and
M. Vergne
V. Baldoni
Affiliation:
Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy (email: baldoni@mat.uniroma2.it)
N. Berline
Affiliation:
Centre de Mathématiques Laurent Schwartz, École Polytechnique, 91128 Palaiseau Cedex, France (email: nicole.berline@math.polytechnique.fr)
M. Köppe
Affiliation:
Department of Mathematics, University of California, Davis, One Shields Avenue, Davis, CA 95616, U.S.A. (email: mkoeppe@math.ucdavis.edu)
M. Vergne
Affiliation:
Université Paris 7 Diderot, Institut Mathématique de Jussieu, 16 rue Clisson, 75013 Paris, France (email: vergne@math.jussieu.fr)
Get access

Abstract

We study intermediate sums, interpolating between integrals and discrete sums, which were introduced by A. Barvinok in [Computing the Ehrhart quasi-polynomial of a rational simplex. Math. Comp. 75 (2006), 1449–1466]. For a given polytope 𝔭 with facets parallel to rational hyperplanes 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 functions. Then we study the Ehrhart theory of these intermediate sums, that is, the dependence of the result as a function of a dilation of the polytope. We provide an algorithm to compute the resulting Ehrhart quasi-polynomials in the form of explicit step-polynomials. These formulas are naturally valid for real (not just integer) dilations and thus provide a direct approach to real Ehrhart theory.

MSC classification

Secondary: 05A15: Exact enumeration problems, generating functions 52C07: Lattices and convex bodies in $n$ dimensions 68R05: Combinatorics 68U05: Computer graphics; computational geometry 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry)
Type
Research Article
Information
Mathematika , Volume 59 , Issue 1 , January 2013 , pp. 1 - 22
Copyright
Copyright © University College London 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Avis, D. and Fukuda, K., Reverse search for enumeration. Discrete Appl. Math. 65(1–3) (1996), 21–46.CrossRefGoogle Scholar
[2]Baldoni, V., Berline, N., De Loera, J. A., Köppe, M. and Vergne, M., Computation of the highest coefficients of weighted Ehrhart quasi-polynomials of rational polyhedra. Found. Comput. Math. Published online 12 November 2011. doi:10.1007/s10208-011-9106-4.CrossRefGoogle Scholar
[3]Baldoni, V., Berline, N., De Loera, J. A., Köppe, M. and Vergne, M., How to integrate a polynomial over a simplex. Math. Comput. 80(273) (2011), 297–325.CrossRefGoogle Scholar
[4]Baldoni, V., Berline, N. and Vergne, M., Local Euler–Maclaurin expansion of Barvinok valuations and Ehrhart coefficients of rational polytopes. In Integer Points in Polyhedra—Geometry, Number Theory, Algebra, Optimization, Statistics (Contemporary Mathematics 452), American Mathematical Society (Providence, RI, 2008), 15–33.Google Scholar
[5]Barvinok, A. I., Polynomial time algorithm for counting integral points in polyhedra when the dimension is fixed. Math. Oper. Res. 19 (1994), 769–779.CrossRefGoogle Scholar
[6]Barvinok, A. I., Computing the Ehrhart quasi-polynomial of a rational simplex. Math. Comp. 75 (2006), 1449–1466.CrossRefGoogle Scholar
[7]Barvinok, A. I. and Pommersheim, J. E., An algorithmic theory of lattice points in polyhedra. In New Perspectives in Algebraic Combinatorics (Mathematical Sciences Research Institute Publications 38) (eds Billera, L. J., Björner, A., Greene, C., Simion, R. E. and Stanley, R. P.), Cambridge University Press (Cambridge, 1999), 91–147.Google Scholar
[8]Berline, N., Köppe, M. and Vergne, M., A discretization-free FPTAS for polynomial optimization over the mixed-integer points in a class of polytopes of varying dimension. Manuscript, in preparation, 2012.Google Scholar
[9]Brion, M., Points entiers dans les polyèdres convexes. Ann. Sci. Éc. Norm. Supér. 21(4) (1988), 653–663.CrossRefGoogle Scholar
[10]Brion, M. and Vergne, M., Residue formulae, vector partition functions and lattice points in rational polytopes. J. Amer. Math. Soc. 10 (1997), 797–833.CrossRefGoogle Scholar
[11]De Loera, J. A., Haws, D., Hemmecke, R., Huggins, P., Tauzer, J. and Yoshida, R., LattE, version 1.2 (2005), http://www.math.ucdavis.edu/∼latte/.CrossRefGoogle Scholar
[12]Köppe, M., A primal Barvinok algorithm based on irrational decompositions. SIAM J. Discrete Math. 21(1) (2007), 220–236.CrossRefGoogle Scholar
[13]Köppe, M. and Verdoolaege, S., Computing parametric rational generating functions with a primal Barvinok algorithm. Electron. J. Combin. 15 (2008), 1–19, #R16.CrossRefGoogle Scholar
[14]Linke, E., Rational Ehrhart quasi-polynomials. J. Combin. Theory Ser. A 118(7) (2011), 1966–1978.CrossRefGoogle Scholar
[15]Verdoolaege, S. and Woods, K. M., Counting with rational generating functions. J. Symbolic Comput. 43(2) (2008), 75–91.CrossRefGoogle Scholar