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Ehrhart quasi-polynomials via Barnes polynomials and discrete moments of parallelepipeds

Sinai Robins

Abstract

We give novel and explicit formulas for the Ehrhart quasi-polynomials of rational simple polytopes, in terms of Barnes polynomials and discrete moments of half-open parallelepipeds. These formulas also hold for all positive dilations of a rational polytope. There is an interesting appearance of an extra complex z-parameter, which seems to allow for more compact formulations. We also give similar formulas for discrete moments of rational polytopes, and their positive dilates, objects known in the literature as sums of polynomials over a polytope. The appearance of the Barnes polynomials and the Barnes numbers allow for explicit computations. From this work, it is clear that the complexity of computing Ehrhart quasi-polynomials lies mainly in the computation of various discrete moments of parallelepipeds. These discrete moments are in general summed over a particular lattice flow on a closed torus, defined in this paper. Some of the consequences involve novel vanishing identities for rational polytopes, novel formulations of Ehrhart polynomials of unimodular polytopes, and a differential equation that extends the work of Eva Linke.

Ehrhart quasi-polynomials via Barnes polynomials and discrete moments of parallelepipeds

Abstract

We give novel and explicit formulas for the Ehrhart quasi-polynomials of rational simple polytopes, in terms of Barnes polynomials and discrete moments of half-open parallelepipeds. These formulas also hold for all positive dilations of a rational polytope. There is an interesting appearance of an extra complex z-parameter, which seems to allow for more compact formulations. We also give similar formulas for discrete moments of rational polytopes, and their positive dilates, objects known in the literature as sums of polynomials over a polytope. The appearance of the Barnes polynomials and the Barnes numbers allow for explicit computations. From this work, it is clear that the complexity of computing Ehrhart quasi-polynomials lies mainly in the computation of various discrete moments of parallelepipeds. These discrete moments are in general summed over a particular lattice flow on a closed torus, defined in this paper. Some of the consequences involve novel vanishing identities for rational polytopes, novel formulations of Ehrhart polynomials of unimodular polytopes, and a differential equation that extends the work of Eva Linke.
Paper Structure (12 sections, 8 theorems, 105 equations, 2 figures)

This paper contains 12 sections, 8 theorems, 105 equations, 2 figures.

Table of Contents

  1. Introduction, and statements of the main results
  2. Known properties of Barnes polynomials
  3. Dimension 1: intervals
  4. Dimension 2: polygons
  5. The unit cube in dimension d
  6. Proof of Theorem \ref{['thm:Ehrhart quasi-polynomial from Barnes']}
  7. Proof of Theorem \ref{['thm:Discrete moments of rational polytopes']}
  8. Proof of Theorem \ref{['thm:MainVanishingIdentities']}: vanishing identities for integer simple polytopes
  9. Proof of Corollary \ref{['Ehrhart for unimodular polytopes']}: Ehrhart for unimodular polytopes
  10. Examples, and some identities for special cases of the Theorems
  11. Special cases of Theorem \ref{['thm:MainVanishingIdentities']}
  12. Further remarks and research directions

Key Result

Theorem 1

Let ${\mathcal{P}} \subset \mathbb{R}^d$ be a $d$-dimensional, simple rational polytope.

Figures (2)

  • Figure 1: Top left: a rational triangle ${\mathcal{P}}$ with a rational vertex $v$. right: the vertex tangent cone $\mathcal{K}_v$ at the vertex $v$. Bottom left: the fundamental parallelepiped $\Pi_v$, with its integer edge vectors $w_1(v), w_2(v)$. Note that $\Pi_v$ always has a vertex at the origin.
  • Figure 2: Left: The fundamental parallelepiped $\Pi_v$ for the vertex $v:= \left(-\frac{1}{2}-\frac{1}{4}\right)$. The lattice flow $\{ LatticeFlow(t) \ \mid \ 0 \leq t \leq \frac{1}{4} \}$ is drawn on the left with red and purple line segments. Each integer point is flowing along its geodesic on the $2$-torus, from $t=0$ until $t = \frac{1}{4}$. Right: the same lattice flow has reached its full closed geodesic flow, at $t=4$.

Theorems & Definitions (31)

  • Definition 1
  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Corollary 1
  • Theorem 2
  • Remark 5
  • Theorem 3
  • ...and 21 more