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Computing parametric weighted Ehrhart polynomials of smooth polytopes

Daniel Hwang, Juliet Whidden, Josephine Yu

TL;DR

This paper shows that when integral polytopes $P_A(\mathbf{b})$ deform with fixed facet normals, the coefficients of weighted Ehrhart and $h^*$-polynomials are piecewise polynomial in $\mathbf{b}$. It develops and implements an algorithm based on the Khovanskii–Pukhlikov weighted Euler–Maclaurin formula to compute these polynomials for smooth polytopes, including type $A$ alcoved polytopes, and provides a catalog of results. It also investigates the signs of weighted $h^*$-coefficients, proving that positivity regions are not universally convex and analyzing the asymptotic behavior under large dilates, where roots converge to those of Eulerian polynomials. The work offers practical parametric tools for computing weighted Ehrhart polynomials and deepens understanding of positivity and root structure in weighted Ehrhart theory, with implications for combinatorial and polyhedral geometry.

Abstract

We show that when integral polytopes are deformed while keeping the same facet normal vectors, the coefficients of weighted Ehrhart and $h^*$-polynomials are piecewise polynomial functions in the ``right hand sides'' of the linear inequalities defining the polytopes. We give an algorithm and an implementation in SageMath for computing these polynomials for smooth polytopes, such as type $A$ alcoved polytopes, using a weighted Euler-Maclaurin type formula by Khovanskiǐ and Pukhlikov. We discuss some natural questions concerning signs of the coefficients of the weighted $h^*$-polynomials.

Computing parametric weighted Ehrhart polynomials of smooth polytopes

TL;DR

This paper shows that when integral polytopes deform with fixed facet normals, the coefficients of weighted Ehrhart and -polynomials are piecewise polynomial in . It develops and implements an algorithm based on the Khovanskii–Pukhlikov weighted Euler–Maclaurin formula to compute these polynomials for smooth polytopes, including type alcoved polytopes, and provides a catalog of results. It also investigates the signs of weighted -coefficients, proving that positivity regions are not universally convex and analyzing the asymptotic behavior under large dilates, where roots converge to those of Eulerian polynomials. The work offers practical parametric tools for computing weighted Ehrhart polynomials and deepens understanding of positivity and root structure in weighted Ehrhart theory, with implications for combinatorial and polyhedral geometry.

Abstract

We show that when integral polytopes are deformed while keeping the same facet normal vectors, the coefficients of weighted Ehrhart and -polynomials are piecewise polynomial functions in the ``right hand sides'' of the linear inequalities defining the polytopes. We give an algorithm and an implementation in SageMath for computing these polynomials for smooth polytopes, such as type alcoved polytopes, using a weighted Euler-Maclaurin type formula by Khovanskiǐ and Pukhlikov. We discuss some natural questions concerning signs of the coefficients of the weighted -polynomials.

Paper Structure

This paper contains 10 sections, 9 theorems, 17 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Piecewise Polynomiality
  3. Algorithms and computations for smooth polytopes
  4. Example: alcoved polytopes
  5. Computational Complexity
  6. Signs of coefficients of weighted $h^{*}$-polynomials
  7. Weighted $h^*$ positivity regions
  8. Sign patterns and linear spaces of $h^{*}$- coefficients
  9. Weighted $h^*$-polynomials of large dilates of polytopes
  10. Acknowledgements

Key Result

Lemma 2.1

For any polynomial weight $w$ of degree $m$, the weighted integer point counting function is a finitely additive measure on $\mathcal{P}$, and it is polynomial of degree $\leq m$.

Figures (3)

  • Figure 1: An example of $P'$ (top right) obtained by a motion of the walls of $P$ (top left). Observe that $P'$'s normal fan (bottom right) is a coarsening of that of $P$ (bottom left).
  • Figure 2: A fixed triangulation of a 2-dimensional alcoved polytope
  • Figure 3: From left to right, the four blue polygons show $P_0,P_1,P_2,P_3$ from Example \ref{['ex:nonconvex']}. The zero locus of the linear weight $w=2y-3x$ is the green line.

Theorems & Definitions (16)

  • Lemma 2.1
  • proof
  • Theorem 2.2: Theorem 1, § 2, Khovanskii-Pukhilkov-Virtual-Polytopes
  • Proposition 2.3
  • Corollary 2.4
  • Definition 3.1
  • Theorem 3.2: Weighted Khovanskiǐ–Pukhlikov
  • proof
  • Theorem 3.3: Corollary 20, how-to-integrate-a-polynomial-over-a-simplex
  • Theorem 3.4
  • ...and 6 more