Generalized Ehrhart polynomials
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- by Sheng Chen, Nan Li and Steven V Sam PDF
- Trans. Amer. Math. Soc. 364 (2012), 551-569 Request permission
Abstract:
Let $P$ be a polytope with rational vertices. A classical theorem of Ehrhart states that the number of lattice points in the dilations $P(n) = nP$ is a quasi-polynomial in $n$. We generalize this theorem by allowing the vertices of $P(n)$ to be arbitrary rational functions in $n$. In this case we prove that the number of lattice points in $P(n)$ is a quasi-polynomial for $n$ 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 $n$, and we explain how these two problems are related.References
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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
- MR Author ID: 836995
- ORCID: 0000-0003-1940-9570
- Email: ssam@math.mit.edu
- 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. - © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 551-569
- MSC (2010): Primary 11D45; Secondary 11D04, 52C07, 05A16
- DOI: https://doi.org/10.1090/S0002-9947-2011-05494-2
- MathSciNet review: 2833591