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 |
| Published: |
Electronic Journal of Combinatorics
2014
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| Online Access: | http://hdl.handle.net/1721.1/89809 |
| Summary: | 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. |
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