Decompositions of rational convex polytopes. (English) Zbl 0812.52012

Stanley, Richard P.

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)

Keywords:

lattice polytope; conjecture of Ehrhart

Citations:

Zbl 0337.10019; Zbl 0335.05010; Zbl 0395.52006