Symmetric Decompositions and Real-Rootedness.

In algebraic, topological, and geometric combinatorics, inequalities among the coefficients of combinatorial polynomials are frequently studied. Recently, a notion called the alternatingly increasing property, which is stronger than unimodality, was introduced. In this paper, we relate the alternatingly increasing property to real-rootedness of the symmetric decomposition of a polynomial to develop a systematic approach for proving the alternatingly increasing property for several classes of polynomials. We apply our results to strengthen and generalize real-rootedness, unimodality, and alternatingly increasing results pertaining to colored Eulerian and derangement polynomials, Ehrhart |$h^\ast$| -polynomials for lattice zonotopes, |$h$| -polynomials of barycentric subdivisions of doubly Cohen–Macaulay level simplicial complexes, and certain local |$h$| -polynomials for subdivisions of simplices. In particular, we prove two conjectures of Athanasiadis. [ABSTRACT FROM AUTHOR]