[Paper Review] Ehrhart polynomial and multiplicity Tutte polynomial

Michele D’Adderio, Luca Moci|arXiv (Cornell University)|Feb 1, 2011

Advanced Combinatorial Mathematics16 references3 citations

TL;DR

This paper establishes a precise algebraic relationship between the Ehrhart polynomial of a zonotope and the multiplicity Tutte polynomial, proving that the Ehrhart polynomial of a zonotope generated by a list of integer vectors is a specialization of the multiplicity Tutte polynomial: $\mathcal{E}_{\mathcal{Z}(X)}(q) = q^n M_X(1 + 1/q, 1)$. This connection provides new combinatorial and geometric interpretations for the coefficients of the Ehrhart polynomial and yields explicit formulae for the volume and number of integer points in the zonotope and its interior.

ABSTRACT

We prove that the Ehrhart polynomial of a zonotope is a specialization of the multiplicity Tutte polynomial. We derive some formulae for the volume and the number of integer points of the zonotope.

Motivation & Objective

• To establish a deep algebraic connection between the Ehrhart polynomial of a zonotope and the multiplicity Tutte polynomial.
• To provide a new interpretation of the coefficients of the Ehrhart polynomial, which remain poorly understood despite known facts about the leading, degree $n-1$, and constant terms.
• To derive explicit formulae for the number of integer points in a zonotope and its interior using the multiplicity Tutte polynomial.
• To strengthen the link between the multiplicity Tutte polynomial and the partition function $\mathcal{P}_X(\lambda)$, particularly through quasipolynomials and toric arrangements.

Proposed method

• The authors define the multiplicity Tutte polynomial $M_X(x,y)$ as a generalization of the classical Tutte polynomial, incorporating lattice arithmetic via the index $m(A) = [\Lambda_A : \langle A\rangle_\mathbb{Z}]$ for subsets $A \subseteq X$.
• They prove that the Ehrhart polynomial of the zonotope $\mathcal{Z}(X)$ is obtained by specializing $M_X(x,y)$ at $x = 1 + 1/q$ and $y = 1$, scaled by $q^n$, i.e., $\mathcal{E}_X(q) = q^n M_X(1 + 1/q, 1)$.
• Two independent proofs are provided: one using a known result from [15] on the number of integer points in $\mathcal{Z}(X)$, and another based on reversing coefficients via $M_X(1+t,1) = t^n \mathcal{E}_X(1/t)$.
• The proof leverages properties of dilated zonotopes, showing $q\mathcal{Z}(X) = \mathcal{Z}(qX)$, and derives a scaling identity for $M_{qX}(x,y)$, which is essential to the main theorem.
• The authors use Ehrhart-Macdonald reciprocity to derive a formula for the number of interior integer points: $\mathcal{I}_X(q) = (-q)^n M_X(1 - 1/q, 1)$.
• They apply these results to compute the volume of $\mathcal{Z}(X)$ as the constant term $M_X(1,1)$, and the number of boundary and interior points via evaluations of $M_X(2,1)$ and $M_X(0,1)$.

Experimental results

Research questions

• RQ1 How can the Ehrhart polynomial of a zonotope be expressed in terms of a known arithmetic polynomial invariant?
• RQ2 What is the precise algebraic relationship between the multiplicity Tutte polynomial and the Ehrhart polynomial of a zonotope?
• RQ3 Can the coefficients of the Ehrhart polynomial, especially those beyond the leading and constant terms, be given a combinatorial or geometric interpretation via the multiplicity Tutte polynomial?
• RQ4 How does the multiplicity Tutte polynomial encode information about the number of integer points in dilated zonotopes and their interiors?
• RQ5 Can the multiplicity Tutte polynomial unify or clarify connections between Ehrhart theory, partition functions, and toric arrangements?

Key findings

• The Ehrhart polynomial of the zonotope $\mathcal{Z}(X)$ is given by $\mathcal{E}_X(q) = q^n M_X(1 + 1/q, 1)$, establishing a direct specialization relationship.
• The number of integer points in $\mathcal{Z}(X)$ is $M_X(2,1)$, which provides a new combinatorial interpretation of this count.
• The volume of the zonotope $\mathcal{Z}(X)$ is equal to $M_X(1,1)$, the constant term of the multiplicity Tutte polynomial.
• The number of integer points in the interior of $\mathcal{Z}(X)$ is $M_X(0,1)$, offering a new formula for the interior point count.
• The number of integer points in the interior of the dilated zonotope $q\mathcal{Z}(X)$ is given by $\mathcal{I}_X(q) = (-q)^n M_X(1 - 1/q, 1)$, derived via Ehrhart-Macdonald reciprocity.
• For the example list $X = \{(3,0),(0,2),(1,1)\}$, the Ehrhart polynomial is $\mathcal{E}_X(q) = 11q^2 + 6q + 1$, the volume is $M_X(1,1) = 11$, and the number of interior points in $\mathcal{Z}(X)$ is $M_X(0,1) = 6$.