# Ehrhart tensor polynomials

## TL;DR

This paper introduces Ehrhart tensor polynomials as a generalization of Ehrhart polynomials, explores their coefficients through triangulations, and investigates positive semidefiniteness properties, extending classical results to tensor settings.

## Contribution

It develops the theory of Ehrhart tensor polynomials, introduces $h^r$-tensor polynomials, and extends Hibi's palindromic theorem to this new framework.

## Key findings

- Coefficients are not monotone with respect to inclusion.
- Positive semidefiniteness proven in dimension two.
- Conjecture of positive semidefiniteness in higher dimensions based on computational evidence.

## Abstract

The notion of Ehrhart tensor polynomials, a natural generalization of the Ehrhart polynomial of a lattice polytope, was recently introduced by Ludwig and Silverstein. We initiate a study of their coefficients. In the vector and matrix cases, we give Pick-type formulas in terms of triangulations of a lattice polygon. As our main tool, we introduce $h^r$-tensor polynomials, extending the notion of the Ehrhart $h^\ast$-polynomial, and, for matrices, investigate their coefficients for positive semidefiniteness. In contrast to the usual $h^\ast$-polynomial, the coefficients are in general not monotone with respect to inclusion. Nevertheless, we are able to prove positive semidefiniteness in dimension two. Based on computational results, we conjecture positive semidefiniteness of the coefficients in higher dimensions. Furthermore, we generalize Hibi's palindromic theorem for reflexive polytopes to $h^r$-tensor polynomials and discuss possible future research directions.