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Transactions of the American Mathematical Society

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Ehrhart polynomials of lattice-face polytopes


Author: Fu Liu
Journal: Trans. Amer. Math. Soc. 360 (2008), 3041-3069
MSC (2000): Primary 05A19; Secondary 52B20
Published electronically: January 8, 2008
MathSciNet review: 2379786
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Abstract | References | Similar Articles | Additional Information

Abstract: There is a simple formula for the Ehrhart polynomial of a cyclic polytope. The purpose of this paper is to show that the same formula holds for a more general class of polytopes, lattice-face polytopes. We develop a way of decomposing any $ d$-dimensional simplex in general position into $ d!$ signed sets, each of which corresponds to a permutation in the symmetric group $ \mathfrak{S}_d,$ and reduce the problem of counting lattice points in a polytope in general position to that of counting lattice points in these special signed sets. Applying this decomposition to a lattice-face simplex, we obtain signed sets with special properties that allow us to count the number of lattice points inside them. We are thus able to conclude the desired formula for the Ehrhart polynomials of lattice-face polytopes.


References [Enhancements On Off] (What's this?)

  • 1. Tom M. Apostol, Introduction to analytic number theory, Springer-Verlag, New York-Heidelberg, 1976. Undergraduate Texts in Mathematics. MR 0434929
  • 2. A. Barvinok, Lattice points, polyhedra, and complexity, Park City Math Institute Lecture Notes (Summer 2004), to appear.
  • 3. M. Beck and S. Robins, Computing the continuous discretely: Integer-point enumeration in polyhedra, Springer (to appear). Preprint at http://math.sfsu.edu/beck/papers/ccd.html.
  • 4. Eugène Ehrhart, Sur les polyèdres rationnels homothétiques à 𝑛 dimensions, C. R. Acad. Sci. Paris 254 (1962), 616–618 (French). MR 0130860
  • 5. Fu Liu, Ehrhart polynomials of cyclic polytopes, J. Combin. Theory Ser. A 111 (2005), no. 1, 111–127. MR 2144858, 10.1016/j.jcta.2004.11.011
  • 6. I. G. Macdonald, Polynomials associated with finite cell-complexes, J. London Math. Soc. (2) 4 (1971), 181–192. MR 0298542

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Additional Information

Fu Liu
Affiliation: Department of Mathematics, Room 2-333, 77 Massachusetts Avenue, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Address at time of publication: Department of Mathematics, University of California, Davis, One Shields Avenue, Davis, California 95616
Email: fuliu@math.mit.edu, fuliu@math.ucdavis.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04288-8
Keywords: Ehrhart polynomial, lattice-face, polytope, signed decomposition
Received by editor(s): February 15, 2006
Received by editor(s) in revised form: March 15, 2006
Published electronically: January 8, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.