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- W4298121931 abstract "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." @default.
- W4298121931 created "2022-10-01" @default.
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- W4298121931 date "2014-11-27" @default.
- W4298121931 modified "2023-09-29" @default.
- W4298121931 title "Local $h$-polynomials, invariants of subdivisions, and mixed Ehrhart theory" @default.
- W4298121931 doi "https://doi.org/10.48550/arxiv.1411.7736" @default.
- W4298121931 hasPublicationYear "2014" @default.
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