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Decompositions of rational convex polytopes. (English) Zbl 0812.52012
Ann. Discrete Math. 6, 333-342 (1980).
Let $$\mathcal P$$ be a rational $$d$$-polytope (its vertices have rational Cartesian coordinates), $$i({\mathcal P},n)$$ the number of lattice points (with integer coordinates) in $$\mathcal P$$, and $$J({\mathcal P},\lambda) = 1 + \sum_{n \geq 1} i({\mathcal P},n)\lambda^ n$$ the corresponding generating function. The function $$J({\mathcal P},\lambda)$$ has been much investigated [see E. Ehrhardt, Polynômes arithmétiques et méthode des polyèdres en combinatoire, Birkhäuser, Basel (1977; Zbl 0337.10019)]; here the author develops further properties. For example, if $$\mathcal P$$ is a lattice polytope, then $$J({\mathcal P},\lambda) = W({\mathcal P},\lambda)/(1 - \lambda)^{d + 1}$$, where $$W({\mathcal P},\lambda)$$ is a polynomial of degree at most $$d$$ with nonnegative integer coefficients (the proof is more geometrical than that of the author [Duke Math. J. 43, No. 3, 511- 531 (1976; Zbl 0335.05010)]); in certain (described) circumstances, these coefficients are simple functions of the numbers of faces of $$\mathcal P$$. In general, $$i({\mathcal P},n)$$ is a near polynomial (“polynôme mixte”) in $$n$$, whose coefficients vary cyclically; the author verifies a conjecture of Ehrhart about when these coefficients are fixed (a proof in a more general situation was given by the reviewer [Arch. Math. 31, 509-516 (1978; Zbl 0395.52006)]).
For the entire collection see [Zbl 0435.00003].

MSC:
 52C22 Tilings in $$n$$ dimensions (aspects of discrete geometry) 05B45 Combinatorial aspects of tessellation and tiling problems 52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
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