Local $h$-polynomials, invariants of subdivisions, and mixed Ehrhart theory

There are natural polynomial invariants of polytopes and lattice polytopes coming from enumerative combinatorics and Ehrhart theory, namely the $h$- and $h^*$-polynomials, respectively. In this paper, we study their generalization to subdivisions and lattice subdivisions of polytopes. By abstracting constructions in mixed Hodge theory, we introduce multivariable polynomials which specialize to the $h$-, $h^*$- polynomials. These polynomials, the mixed $h$-polynomial and the (refined) limit mixed $h^*$-polynomial have rich symmetry, non-negativity, and unimodality properties, which both refine known properties of the classical polynomials, and reveal new structure. For example, we prove a lower bound theorem for a related invariant called the local $h^*$-polynomial. We introduce our polynomials by developing a very general formalism for studying subdivisions of Eulerian posets that extends the work of Stanley, Brenti and Athanasiadis on local $h$-vectors. In particular, we prove a conjecture of Nill and Schepers, and answer a question of Athanasiadis.