Rational Ehrhart quasi-polynomials

Linke, Eva

doi:10.1016/j.jcta.2011.03.007

Cited by 27 publications

(17 citation statements)

References 14 publications

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“…The rational polytope P G , associated to a {1, 3}-graph G, enjoys some fascinating symmetry. Linke [9] considered the extension of L P (t) for all nonnegative real numbers t. Royer [13] defined a polytope to be semi-reflexive if L P (s) = L P (⌊s⌋) for every nonnegative real number s. One can verify that P G is semi-reflexive. A polytope is reflexive if it is integral, the origin is in its interior, and it is semi-reflexive [2].…”

Section: Questionmentioning

confidence: 99%

Cubic Graphs, Their Ehrhart Quasi-Polynomials, and a Scissors Congruence Phenomenon

Fernandes

Pina

Alfonsín

et al. 2020

Discrete Comput Geom

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The scissors congruence conjecture for the unimodular group is an analogue of Hilbert's third problem, for the equidecomposability of polytopes. Liu and Osserman studied the Ehrhart quasi-polynomials of polytopes naturally associated to graphs whose vertices have degree one or three. In this paper, we prove the scissors congruence conjecture, posed by Haase and McAllister, for this class of polytopes.The key ingredient in the proofs is the nearest neighbor interchange on graphs and a naturally arising piecewise unimodular transformation. * FAPESP 13/03447-6 and 15/10323-7, CNPq 456792/2014-7 and 452507/2016-2, CAPES PROEX.

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Section: Questionmentioning

confidence: 99%

Cubic Graphs, Their Ehrhart Quasi-Polynomials, and a Scissors Congruence Phenomenon

Fernandes

Pina

Alfonsín

et al. 2020

Discrete Comput Geom

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“…, c n are periodic functions in t. Ehrhart-Macdonald reciprocity carries over verbatim to the rational case. Further yet, very recent results [1,2,21] extended Ehrhart (quasi-)polynomials by allowing rational or real dilation factors when counting lattice points in rational polytopes. We finish this section by mentioning that there are alternative ways of proving Theorem 2, see, e.g., [6,Chapter 4] and [25]; our proof followed Ehrhart's original lines [14] (Section 3.1) and [28, Chapter 4] (Section 3.2).…”

Section: Latticementioning

confidence: 99%

Combinatorial Reciprocity Theorems

Beck

2018

Graduate Studies in Mathematics

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A common theme of enumerative combinatorics is formed by counting functions that are polynomials evaluated at positive integers. In this expository paper, we focus on four families of such counting functions connected to hyperplane arrangements, lattice points in polyhedra, proper colorings of graphs, and P-partitions. We will see that in each instance we get interesting information out of a counting function when we evaluate it at a negative integer (and so, a priori the counting function does not make sense at this number). Our goals are to convey some of the charm these "alternative" evaluations of counting functions exhibit, and to weave a unifying thread through various combinatorial reciprocity theorems by looking at them through the lens of geometry, which will include some scenic detours through other combinatorial concepts.Date: 30 December 2011.

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“…The results remains valid for non-negative real values of t. Thus our method provides a direct proof of a real Ehrhart theorem (Theorem 31). This extension of Ehrhart theory to real dilation factors has been studied by E. Linke [13] for the classical case. It is even more natural for the intermediate valuations, as one of the cases is L = V .…”

Section: Introductionmentioning

confidence: 96%

Intermediate Sums on Polyhedra: Computation and Real Ehrhart Theory

et al. 2012

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We study intermediate sums, interpolating between integrals and discrete sums, which were introduced by A. Barvi-nok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), 1449–1466]. For a given semi-rational polytope p and a rational subspace L, we integrate a given polyno-mial function h over all lattice slices of the polytope p parallel to the subspace L and sum up the integrals. We first develop an al-gorithmic theory of parametric intermediate generating functions. Then we study the Ehrhart theory of these intermediate sums, that is, the dependence of the result as a function of a dilation of the polytope. We provide an algorithm to compute the resulting Ehrhart quasi-polynomials in the form of explicit step polynomi-als. These formulas are naturally valid for real (not just integer) dilations and thus provide a direct approach to real Ehrhart theory

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Rational Ehrhart quasi-polynomials

Cited by 27 publications

References 14 publications

Cubic Graphs, Their Ehrhart Quasi-Polynomials, and a Scissors Congruence Phenomenon

Cubic Graphs, Their Ehrhart Quasi-Polynomials, and a Scissors Congruence Phenomenon

Combinatorial Reciprocity Theorems

Intermediate Sums on Polyhedra: Computation and Real Ehrhart Theory

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