Generalized Ehrhart polynomials

Authors:
Sheng Chen, Nan Li and Steven V Sam

Journal:
Trans. Amer. Math. Soc. **364** (2012), 551-569

MSC (2010):
Primary 11D45; Secondary 11D04, 52C07, 05A16

Published electronically:
June 29, 2011

MathSciNet review:
2833591

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a polytope with rational vertices. A classical theorem of Ehrhart states that the number of lattice points in the dilations is a quasi-polynomial in . We generalize this theorem by allowing the vertices of to be arbitrary rational functions in . In this case we prove that the number of lattice points in is a quasi-polynomial for sufficiently large. Our work was motivated by a conjecture of Ehrhart on the number of solutions to parametrized linear Diophantine equations whose coefficients are polynomials in , and we explain how these two problems are related.

**[Bar]**Alexander Barvinok,*Integer points in polyhedra*, Zurich Lectures in Advanced Mathematics, European Mathematical Society (EMS), Zürich, 2008. MR**2455889****[BR]**Matthias Beck and Sinai Robins,*Computing the continuous discretely*, Undergraduate Texts in Mathematics, Springer, New York, 2007. Integer-point enumeration in polyhedra. MR**2271992****[Ehr]**E. Ehrhart,*Polynômes arithmétiques et méthode des polyèdres en combinatoire*, Birkhäuser Verlag, Basel-Stuttgart, 1977. International Series of Numerical Mathematics, Vol. 35. MR**0432556****[New]**Morris Newman,*Integral matrices*, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 45. MR**0340283****[Sta]**Richard P. Stanley,*Enumerative combinatorics. Vol. 1*, Cambridge Studies in Advanced Mathematics, vol. 49, Cambridge University Press, Cambridge, 1997. With a foreword by Gian-Carlo Rota; Corrected reprint of the 1986 original. MR**1442260****[Xu]**Zhiqiang Xu,*An explicit formulation for two dimensional vector partition functions*, Integer points in polyhedra—geometry, number theory, representation theory, algebra, optimization, statistics, Contemp. Math., vol. 452, Amer. Math. Soc., Providence, RI, 2008, pp. 163–178. MR**2405771**, 10.1090/conm/452/08781

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Additional Information

**Sheng Chen**

Affiliation:
Department of Mathematics, Harbin Institute of Technology, Harbin, People’s Republic of China 150001

Email:
schen@hit.edu.cn

**Nan Li**

Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Email:
nan@math.mit.edu

**Steven V Sam**

Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Email:
ssam@math.mit.edu

DOI:
https://doi.org/10.1090/S0002-9947-2011-05494-2

Keywords:
Ehrhart polynomials,
Diophantine equations,
lattice points,
polytopes,
quasi-polynomials

Received by editor(s):
May 14, 2010

Received by editor(s) in revised form:
September 6, 2010, October 8, 2010, and October 26, 2010

Published electronically:
June 29, 2011

Additional Notes:
The first author was sponsored by Project 11001064 supported by the National Natural Science Foundation of China, and Project HITC200701 supported by the Science Research Foundation in Harbin Institute of Technology.

The third author was supported by an NSF graduate fellowship and an NDSEG fellowship.

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.