Transactions of the American Mathematical Society

Author(s): A. Stapledon
Journal: Trans. Amer. Math. Soc. 361 (2009), 5615-5626.
MSC (2000): Primary 52B20
Posted: May 13, 2009
Retrieve article in: PDF

Abstract: For any lattice polytope

, we consider an associated polynomial $\bar{\delta}_{P}(t)$ and describe its decomposition into a sum of two polynomials satisfying certain symmetry conditions. As a consequence, we improve upon known inequalities satisfied by the coefficients of the Ehrhart $\delta$ -vector of a lattice polytope. We also provide combinatorial proofs of two results of Stanley that were previously established using techniques from commutative algebra. Finally, we give a necessary numerical criterion for the existence of a regular unimodular lattice triangulation of the boundary of a lattice polytope.

1.: Christos A. Athanasiadis, $h\sp \ast$ -vectors, Eulerian polynomials and stable polytopes of graphs, Electron. J. Combin. 11 (2004/06), no. 2, Research Paper 6, 13 pp. (electronic). MR 2120101 (2006a:05170)
2.: -, Ehrhart polynomials, simplicial polytopes, magic squares and a conjecture of Stanley, J. Reine Angew. Math. 583 (2005), 163-174. MR 2146855 (2006a:05171)
3.: M. Beck, J. A. De Loera, M. Develin, J. Pfeifle, and R. P. Stanley, Coefficients and roots of Ehrhart polynomials, Integer points in polyhedra--geometry, number theory, algebra, optimization, Contemp. Math., vol. 374, Amer. Math. Soc., Providence, RI, 2005, pp. 15-36. MR 2134759 (2006e:52019)
4.: U. Betke and P. McMullen, Lattice points in lattice polytopes, Monatsh. Math. 99 (1985), no. 4, 253-265. MR 799674 (87e:52019)
5.: E. Ehrhart, Sur un problème de géométrie diophantienne linéaire. I. Polyèdres et réseaux, J. Reine Angew. Math. 226 (1967), 1-29. MR 0213320 (35:4184)
6.: -, Sur un problème de géométrie diophantienne linéaire. II. Systèmes diophantiens linéaires, J. Reine Angew. Math. 227 (1967), 25-49. MR 0217010 (36:105)
7.: William Fulton, Introduction to toric varieties, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993, The William H. Roever Lectures in Geometry. MR 1234037 (94g:14028)
8.: Martin Henk and Makoto Tagami, Lower bounds on the coefficients of Ehrhart polynomials, European J. Combin. 30 (2009), no. 1, 70-83. MR 2460218
9.: Takayuki Hibi, Some results on Ehrhart polynomials of convex polytopes, Discrete Math. 83 (1990), no. 1, 119-121. MR 1065691 (91g:52008)
10.: -, Ehrhart polynomials of convex polytopes, -vectors of simplicial complexes, and nonsingular projective toric varieties, Discrete and computational geometry (New Brunswick, NJ, 1989/1990), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 6, Amer. Math. Soc., Providence, RI, 1991, pp. 165-177. MR 1143294 (92j:52018)
11.: -, Algebraic combinatorics of convex polytopes, Carslaw Publications, Australia, 1992.
12.: -, Dual polytopes of rational convex polytopes, Combinatorica 12 (1992), no. 2, 237-240. MR 1179260 (93f:52018)
13.: -, A lower bound theorem for Ehrhart polynomials of convex polytopes, Adv. Math. 105 (1994), no. 2, 162-165. MR 1275662 (95b:52018)
14.: -, Star-shaped complexes and Ehrhart polynomials, Proc. Amer. Math. Soc. 123 (1995), no. 3, 723-726. MR 1249883 (95d:52012)
15.: Mircea Mustaţa and Sam Payne, Ehrhart polynomials and stringy Betti numbers, Math. Ann. 333 (2005), no. 4, 787-795. MR 2195143 (2007c:14055)
16.: Sam Payne, Ehrhart series and lattice triangulations, Discrete Comput. Geom. 40 (2008), no. 3, 365-376. MR 2443289
17.: Richard P. Stanley, Hilbert functions of graded algebras, Advances in Math. 28 (1978), no. 1, 57-83. MR 0485835 (58:5637)
18.: -, Decompositions of rational convex polytopes, Ann. Discrete Math. 6 (1980), 333-342, Combinatorial mathematics, optimal designs and their applications (Proc. Sympos. Combin. Math. and Optimal Design, Colorado State Univ., Fort Collins, Colo., 1978). MR 593545 (82a:52007)
19.: -, The number of faces of a simplicial convex polytope, Adv. in Math. 35 (1980), no. 3, 236-238. MR 563925 (81f:52014)
20.: -, On the Hilbert function of a graded Cohen-Macaulay domain, J. Pure Appl. Algebra 73 (1991), no. 3, 307-314. MR 1124790 (92f:13017)
21.: Alan Stapledon, Weighted Ehrhart theory and orbifold cohomology, Adv. Math. 219 (2008), no. 1, 63-88. MR 2435420

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 52B20

A. Stapledon
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: astapldn@umich.edu

DOI: 10.1090/S0002-9947-09-04776-X
PII: S 0002-9947(09)04776-X
Received by editor(s): January 9, 2008
Received by editor(s) in revised form: February 22, 2008
Posted: May 13, 2009
Additional Notes: The author was supported by Mircea Mustata's Packard Fellowship and by an Eleanor Sophia Wood travelling scholarship from the University of Sydney
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS