Multivariate volume, Ehrhart, and-polynomials of polytropes

Marie-Charlotte Brandenburg, Sophia Elia, Leon Zhang

Abstract: The univariate Ehrhart and h⁎<math><msup is="true"><mrow is="true"><mi is="true">h</mi></mrow><mrow is="true"><mo is="true">⁎</mo></mrow></msup></math>-polynomials of lattice polytopes have been widely studied. We describe methods from toric geometry for computing multivariate versions of volume, Ehrhart and h⁎<math><msup is="true"><mrow is="true"><mi is="true">h</mi></mrow><mrow is="true"><mo is="true">⁎</mo></mrow></msup></math>-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<math><mn is="true">2</mn><mo is="true">,</mo><mn is="true">3</mn></math> 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.