Multivariate volume, Ehrhart, and h⁎-polynomials of polytropes. (January 2023) - Brandenburg, Marie-Charlotte; Elia, Sophia; Zhang, Leon
Abstract: The univariate Ehrhart and h ⁎ -polynomials of lattice polytopes have been widely studied. We describe methods from toric geometry for computing multivariate versions of volume, Ehrhart and h ⁎ -polynomials of lattice polytropes, which are both tropically and classically convex, and are also known as alcoved polytopes of type A . These algorithms are applied to all polytropes of dimensions 2, 3 and 4, yielding a large class of integer polynomials. We give a complete combinatorial description of the coefficients of volume polynomials of 3-dimensional polytropes in terms of regular central subdivisions of the fundamental polytope. Finally, we provide a partial characterization of the analogous coefficients in dimension 4.