Symmetric Decompositions and Real-Rootedness

By Petter Brand'en, Liam Solus. International mathematics research notices, 2018

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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.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics

View Paper: https://www.semanticscholar.org/paper/91453d18f65ebd12a81f4b8ecd8ee073f9a7f35d