×

Some results on Ehrhart polynomials of convex polytopes. (English) Zbl 0708.52005

In this short communication the author states some recent results concerning the \(\delta\)-vectors of d-dimensional integral convex polytopes. He gives a linear inequality for the components of the \(\delta\)-vector, and he characterizes the case that the \(\delta\)-vector is symmetric. For special polytopes he explicitly computes the Ehrhart polynomials. The proofs will be published in forthcoming papers.
Reviewer: J.Müller

MSC:

52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
52B11 \(n\)-dimensional polytopes
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ehrhart, E., Polynômes Arithmétiques et Méthode des Polyèdres en Combinatoire (1977), Birkhäuser: Birkhäuser Basel/Stuttgart · Zbl 0337.10019
[2] Grünbaum, B., Convex Polytopes (1967), John Wiley: John Wiley New York · Zbl 0163.16603
[3] T. Hibi, Dual polytopes of rational convex polytopes, submitted.; T. Hibi, Dual polytopes of rational convex polytopes, submitted. · Zbl 0758.52009
[4] T. Hibi, A combinatorial self-reciprocity theorem for Ehrhart quasi-polynomials of rational convex polytopes, submitted.; T. Hibi, A combinatorial self-reciprocity theorem for Ehrhart quasi-polynomials of rational convex polytopes, submitted. · Zbl 0807.52011
[5] T. Hibi, Toroidal posets, preprint.; T. Hibi, Toroidal posets, preprint.
[6] T. Hibi, The Ehrhart polynomial of a convex polytope, in preparation.; T. Hibi, The Ehrhart polynomial of a convex polytope, in preparation. · Zbl 0807.52011
[7] Hochster, M., Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes, Ann. Math., 96, 318-337 (1972) · Zbl 0233.14010
[8] Stanley, R., Hilbert functions of graded algebras, Adv. in Math., 28, 57-83 (1978) · Zbl 0384.13012
[9] Stanley, R., Decompositions of rational convex polytopes, Ann. Discrete Math., 6, 333-342 (1980) · Zbl 0812.52012
[10] Stanley, R., Two poset polytopes, Discrete Comput. Geom., 1, 9-23 (1986) · Zbl 0595.52008
[11] Stanley, R., Enumerative Combinatorics, Volume 1 (1986), Wadsworth: Wadsworth Monterey, CA · Zbl 0608.05001
[12] Stanley, R., On the Hilbert function of a graded Cohen-Macaulay domain (February, 1990), preprint
[13] Hensley, D., Lattice vertex polytopes with interior lattice points, Pacific J. Math., 105, 183-191 (1983) · Zbl 0471.52006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.