Rational Ehrhart quasi-polynomials

Linke, Eva

doi:10.48550/arxiv.1006.5612

Cited by 5 publications

(9 citation statements)

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“…Next, we give a third quick corollary of the main result, this time retrieving a reciprocity law for the solid angle polynomial of a real polytope, which appeared in [19]. Looking back at equation (22) allows us to extend the domain of the solid-angle sum to all nonzero real numbers t ∈ R. In other words, we can extend the function A P (t), already defined for all positive reals t in (22), to include all nonzero t ∈ R: (35) A P (t) := t d lim…”

Section: Retrieving Classical Results From the Main Theoremmentioning

confidence: 99%

Fourier transforms of polytopes, solid angle sums, and discrete volume

Diaz¹,

Le²,

Robins³

2016

Preprint

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Given a real closed polytope P , we first describe the Fourier transform of its indicator function by using iterations of Stokes' theorem. We then use the ensuing Fourier transform formulations, together with the Poisson summation formula, to give a new algorithm to count fractionally-weighted lattice points inside the one-parameter family of all real dilates of P . The combinatorics of the face poset of P plays a central role in the description of the Fourier transform of P . We also obtain a closed form for the codimension-1 coefficient that appears in an expansion of this sum in powers of the real dilation parameter t. This closed form generalizes some known results about the Macdonald solid-angle polynomial, which is the analogous expression traditionally obtained by requiring that t assumes only integer values. Although most of the present methodology applies to all real polytopes, a particularly nice application is to the study of all real dilates of integer (and rational) polytopes.

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Section: Retrieving Classical Results From the Main Theoremmentioning

confidence: 99%

Fourier transforms of polytopes, solid angle sums, and discrete volume

Diaz¹,

Le²,

Robins³

2016

Preprint

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“…When p is a lattice polytope, then the coefficients E m (p, h; t) do not depend on t, and t → S(tp, h) becomes a polynomial. E. Linke [13] has proved that the expression ( 13) is still valid for real dilations t ∈ R. This is the generalized setting that we are going to use in this section. Note that if we allow real dilations, then even for lattice polytopes p we obtain a quasi-polynomial rather than a polynomial; the fractional part function {•} q with q = 1 appears in the expressions for the coefficients E m (p, h; t).…”

Section: Ehrhart Quasipolynomials For Intermediate Valuationsmentioning

confidence: 90%

“…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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“…However, as P is a rational polytope, it suffices to compute rehr(P; λ) at certain rational arguments to fully understand rehr(P; λ); we will (quantify and) make this statement precise shortly. To the best of our knowledge, Linke [14] initiated the study of rehr(P; λ) from the Ehrhart viewpoint. She proved several fundamental results starting with the fact that rehr(P; λ) is a quasipolynomial in the real variable λ, that is,…”

Section: Introductionmentioning

confidence: 99%

“…Our goal is to add a generating-function viewpoint to [1,14], one that is inspired by [17]. To set it up, we need to make a definition.…”

Section: Introductionmentioning

confidence: 99%

Rational Ehrhart Theory

Beck¹,

Sophia²,

Rehberg³

2021

Preprint

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The Ehrhart quasipolynomial of a rational polytope P encodes fundamental arithmetic data of P, namely, the number of integer lattice points in positive integral dilates of P. Ehrhart quasipolynomials were introduced in the 1960s, satisfy several fundamental structural results and have applications in many areas of mathematics and beyond. The enumerative theory of lattice points in rational (equivalently, real) dilates of rational polytopes is much younger, starting with work by Linke (2011), Baldoni-Berline-Koeppe-Vergne (2013), and Stapledon (2017). We introduce a generating-function ansatz for rational Ehrhart quasipolynomials, which unifies several known results in classical and rational Ehrhart theory. In particular, we define γ-rational Gorenstein polytopes, which extend the classical notion to the rational setting and encompass the generalized reflexive polytopes studied by Fiset-Kasprzyk (2008) and Kasprzyk-Nill (2012).

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Rational Ehrhart quasi-polynomials

Cited by 5 publications

References 0 publications

Fourier transforms of polytopes, solid angle sums, and discrete volume

Fourier transforms of polytopes, solid angle sums, and discrete volume

Intermediate Sums on Polyhedra: Computation and Real Ehrhart Theory

Rational Ehrhart Theory

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