Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 39, No. 1, pp. 205 - 218 (1998)

Triangulations of Integral Polytopes and Ehrhart Polynomials

Jean-Michel Kantor

Institut de Mathematiques, Universite Pierre et Marie Curie
Case 247, 4, Place Jussieu, F-75525 Paris Cedex 05

Abstract: We are interested in polytopes in real space of arbitrary dimension, having vertices with integral coordinates: integral polytopes. The recent increase of interest for the study of these polytopes and their triangulations has various motivations; let us mention the main ones: \item{$\bullet$} the beautiful theory of toric varieties has built a bridge between algebraic geometry and the combinatorics of these integral polytopes [F]. Triangulations of cones and polytopes occur naturally for example in problems of existence of crepant resolution of singularities [B], [D]. \item{$\bullet$} The work of the school of I. M. Gelfand on secondary polytopes gives a new insight on triangulations, with applications to algebraic geometry and group theory [G]. \item{$\bullet$} In statistical physics, random tilings lead to some interesting problems dealing with triangulations of order polytopes [M].

With these motivations in mind, we introduce new tools which might bring some informations: Generalizations of the Ehrhart polynomial (counting points modulo congruence), discrete length between integral points and studying the geometry associated to it, arithmetic Euler-Poincare formula which gives, in dimension 3, the Ehrhart polynomial in terms of the $f$-vector of a minimal triangulation of the polytope (Theorem 6).

Finally, let us mention the results in dimension 2 of the late Peter Greenberg which led us to the study of "Arithmetical PL-topology" which, we believe with M. Gromov, D. Sullivan, and P. Vogel, has not yet revealed all its beauties. We thank these mathematicians for their interest, and Professor Ito for his kind invitation to The Seminar at R.I.M.S. in October 1995 where part of these results were given. \item{[B]} Batyrev, V.: Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties. J. Alg. Geom., 3 (1994), 493-535. \item{[D]} Dais, D.: Enumerative combinatorics of invariants of certain complex threefolds with trivial canonical bundle. Bonner Math. Schriften, 279 (1995). \item{[M]} Destainville-Mosseri; Bailly, F.: Entropy in random tilings: a geometrical analysis in configuration space. ICQ5, World Scientific, 1995, Proceedings of Avignon Conference. \item{[F]} Fulton, W.: Introduction to toric varieties, Princeton Univ. Press. \item{[G]} Gelfand, I. M.; Kapranov; Zelevinsky: Discriminants, resultants and multidimensional determinants, Birkhäuser.

Keywords: integral polytopes; toric varieties; Ehrhart polynomial; Arithmetical PL-topology

Classification (MSC91): 52B12

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