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].

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