Ehrhart tensor polynomials

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.