A Finite Calculus Approach to Ehrhart Polynomials
A rational polytope is the convex hull of a finite set of points in R[superscript d] with rational coordinates. Given a rational polytope P⊆R[superscript d], Ehrhart proved that, for t∈Z≥[subscript 0[, the function #(tP∩Z[superscript d]) agrees with a quasi-polynomial L[subscript P](t), called the E...
| Main Authors: | , |
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| Other Authors: | |
| Format: | Article |
| Language: | en_US |
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Electronic Journal of Combinatorics
2014
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| Online Access: | http://hdl.handle.net/1721.1/89809 |
| _version_ | 1811082282758307840 |
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| author | Sam, Steven V. Woods, Kevin M. |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Sam, Steven V. Woods, Kevin M. |
| author_sort | Sam, Steven V. |
| collection | MIT |
| description | A rational polytope is the convex hull of a finite set of points in R[superscript d] with rational coordinates. Given a rational polytope P⊆R[superscript d], Ehrhart proved that, for t∈Z≥[subscript 0[, the function #(tP∩Z[superscript d]) agrees with a quasi-polynomial L[subscript P](t), called the Ehrhart quasi-polynomial. The Ehrhart quasi-polynomial can be regarded as a discrete version of the volume of a polytope. We use that analogy to derive a new proof of Ehrhart's theorem. This proof also allows us to quickly prove two other facts about Ehrhart quasi-polynomials: McMullen's theorem about the periodicity of the individual coefficients of the quasi-polynomial and the Ehrhart–Macdonald theorem on reciprocity. |
| first_indexed | 2024-09-23T12:00:38Z |
| format | Article |
| id | mit-1721.1/89809 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T12:00:38Z |
| publishDate | 2014 |
| publisher | Electronic Journal of Combinatorics |
| record_format | dspace |
| spelling | mit-1721.1/898092022-09-27T23:27:08Z A Finite Calculus Approach to Ehrhart Polynomials Sam, Steven V. Woods, Kevin M. Massachusetts Institute of Technology. Department of Mathematics Sam, Steven V. A rational polytope is the convex hull of a finite set of points in R[superscript d] with rational coordinates. Given a rational polytope P⊆R[superscript d], Ehrhart proved that, for t∈Z≥[subscript 0[, the function #(tP∩Z[superscript d]) agrees with a quasi-polynomial L[subscript P](t), called the Ehrhart quasi-polynomial. The Ehrhart quasi-polynomial can be regarded as a discrete version of the volume of a polytope. We use that analogy to derive a new proof of Ehrhart's theorem. This proof also allows us to quickly prove two other facts about Ehrhart quasi-polynomials: McMullen's theorem about the periodicity of the individual coefficients of the quasi-polynomial and the Ehrhart–Macdonald theorem on reciprocity. 2014-09-18T16:50:29Z 2014-09-18T16:50:29Z 2010-04 2009-11 Article http://purl.org/eprint/type/JournalArticle 1077-8926 http://hdl.handle.net/1721.1/89809 Sam, Steven V., and Kevin M. Woods. "A Finite Calculus Approach to Ehrhart Polynomials." Electronic Journal of Combinatorics, Volume 17 (2010). en_US http://www.combinatorics.org/ojs/index.php/eljc/article/view/v17i1r68 Electronic Journal of Combinatorics Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf Electronic Journal of Combinatorics Electronic Journal of Combinatorics |
| spellingShingle | Sam, Steven V. Woods, Kevin M. A Finite Calculus Approach to Ehrhart Polynomials |
| title | A Finite Calculus Approach to Ehrhart Polynomials |
| title_full | A Finite Calculus Approach to Ehrhart Polynomials |
| title_fullStr | A Finite Calculus Approach to Ehrhart Polynomials |
| title_full_unstemmed | A Finite Calculus Approach to Ehrhart Polynomials |
| title_short | A Finite Calculus Approach to Ehrhart Polynomials |
| title_sort | finite calculus approach to ehrhart polynomials |
| url | http://hdl.handle.net/1721.1/89809 |
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