Multivariate volume, Ehrhart, and $h^{\ast }$-polynomials of polytropes

The univariate Ehrhart and $h^{\ast }$-polynomials of lattice polytopes have been widely studied. We describe methods from toric geometry for computing multivariate versions of volume, Ehrhart and $h^{\ast }$-polynomials of lattice polytropes, which are both tropically and classically convex. 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.