The minimum period of the Ehrhart quasi-polynomial of a rational polytope

McAllister, Tyrrell B.; Woods, Kevin M.

doi:10.1016/j.jcta.2004.08.006

Cited by 44 publications

(22 citation statements)

References 2 publications

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“…When P is a simplex, we deduce a formula for the generating series of f Pg (m) (Proposition 6.1). A pseudo-integral polytope is a rational polytope whose Ehrhart quasi-polynomial is a polynomial (see the work of De Loera, McAllister and Woods [10,24]), and we apply this formula to construct new pseudo-integral polytopes in all dimensions in Section 10.…”

Section: Introductionmentioning

confidence: 99%

Equivariant Ehrhart theory

Stapledon

2011

Advances in Mathematics

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Motivated by representation theory and geometry, we introduce and develop an equivariant generalization of Ehrhart theory, the study of lattice points in dilations of lattice polytopes. We prove representation-theoretic analogues of numerous classical results, and give applications to the Ehrhart theory of rational polytopes and centrally symmetric polytopes. We also recover a character formula of Procesi, Dolgachev, Lunts and Stembridge for the action of a Weyl group on the cohomology of a toric variety associated to a root system.

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Section: Introductionmentioning

confidence: 99%

Equivariant Ehrhart theory

Stapledon

2011

Advances in Mathematics

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“…This phenomenon is called period collapse and has been subject to active research in the last years. McAllister and Woods [11] studied the 1-and 2-dimensional case with the result that period collapse does not occur in dimension 1, and they gave a characterization of those rational polygons in dimension 2 whose Ehrhart quasi-polynomial is a polynomial. They also showed that the minimal periods of G i (P , ·) are not necessarily decreasing with i.…”

Section: Introductionmentioning

confidence: 99%

Rational Ehrhart quasi-polynomials

Linke

2011

Journal of Combinatorial Theory, Series A

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Ehrhart's famous theorem states that the number of integral points in a rational polytope is a quasi-polynomial in the integral dilation factor. We study the case of rational dilation factors. It turns out that the number of integral points can still be written as a rational quasi-polynomial. Furthermore, the coefficients of this rational quasi-polynomial are piecewise polynomial functions and related to each other by derivation.

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“…Quasi-period collapse cannot happen in dimension 1, but there exist families of polygons in R 2 with arbitrarily large denominators whose minimum quasi-periods are 1. This result was originally proved in [MW05], where the proof of polynomiality involved subdividing the polygons into polygonal pieces whose Ehrhart quasi-polynomials could be computed. The periodic parts for these pieces could be seen by inspection to cancel, with the result that the counting function for the entire polygon was a polynomial.…”

Section: Introductionmentioning

confidence: 99%

Quasi-period collapse and 𝐺𝐿_{𝑛}(ℤ)-scissors congruence in rational polytopes

Haase¹,

McAllister²

2008

Contemporary Mathematics

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Quasi-period collapse occurs when the Ehrhart quasi-polynomial of a rational polytope has a quasi-period less than the denominator of that polytope. This phenomenon is poorly understood, and all known cases in which it occurs have been proven with ad hoc methods. In this note, we present a conjectural explanation for quasi-period collapse in rational polytopes. We show that this explanation applies to some previous cases appearing in the literature. We also exhibit examples of Ehrhart polynomials of rational polytopes that are not the Ehrhart polynomials of any integral polytope.Our approach depends on the invariance of the Ehrhart quasi-polynomial under the action of affine unimodular transformations. Motivated by the similarity of this idea to the scissors congruence problem, we explore the development of a Dehn-like invariant for rational polytopes in the lattice setting.

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The minimum period of the Ehrhart quasi-polynomial of a rational polytope

Cited by 44 publications

References 2 publications

Equivariant Ehrhart theory

Equivariant Ehrhart theory

Rational Ehrhart quasi-polynomials

Quasi-period collapse and 𝐺𝐿_{𝑛}(ℤ)-scissors congruence in rational polytopes

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