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Lattice points in convex polytopes. (Points entiers dans les polytopes convexes.) (French) Zbl 0847.52015

Séminaire Bourbaki. Volume 1993/94. Exposés 775-789. Paris: Société Mathématique de France, Astérisque. 227, 145-169 (Exp. No. 780) (1995).
Let \(M\) be a lattice in a \(d\)-dimensional real vector space (there is no harm in thinking of \(M\) as the usual integer lattice). An integer or lattice (convex) polytope \(P\) is one whose vertices lie in \(M\). It is well-known [see, for example, E. Ehrhart, C. R. Acad. Sci., Paris, Ser. A 265, 5-7 (1967; Zbl 0147.30701)] that \(i_P(n):=\text{card} (M \cap nP)\) is a polynomial in \(n\) of degree \(d\), say \(i_P (n) = a_0 (P) + a_1 ( P )n+\cdots+a_d(P)n^d\). For practical as well as theoretical reasons, there is much interest in determining the coefficients \(a_j(P)\). A few are easily found: \(a_0(P) = 1\) (the Euler characteristic), \(a_d (P)\) is just the volume (relative to \(M)\), and \(a_{d - 1} (P)\) is half the lattice surface area. A wide range of attacks has been made on the general problem, for example, L. J. Mordell [J. Indian Math. Soc., n. Ser. 15, 41-46 (1951; Zbl 0043.05101)] using Dedekind sums for tetrahedra.
In recent years, some of the more striking have come from the connexion between lattice polytopes and toric varieties. In particular, considerable success has resulted from the use of Todd classes and equivariant \(K\)-theory.
In this article, the author surveys much of the current progress, detailing the relationships between the different approaches.
For the entire collection see [Zbl 0811.00012].

MSC:

52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
11-02 Research exposition (monographs, survey articles) pertaining to number theory
52-02 Research exposition (monographs, survey articles) pertaining to convex and discrete geometry
11H06 Lattices and convex bodies (number-theoretic aspects)
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