Symmetric Decompositions and Real-Rootedness
| Autor: | Petter Brändén, Liam Solus |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Pure mathematics
Polynomial Mathematics::Combinatorics General Mathematics 010102 general mathematics Eulerian path 0102 computer and information sciences Barycentric coordinate system Lattice (discrete subgroup) 01 natural sciences Unimodality symbols.namesake Derangement 010201 computation theory & mathematics FOS: Mathematics symbols Mathematics - Combinatorics Combinatorics (math.CO) 0101 mathematics Geometric combinatorics Algebraic number Mathematics |
| Zdroj: | International Mathematics Research Notices. 2021:7764-7798 |
| ISSN: | 1687-0247 1073-7928 |
| DOI: | 10.1093/imrn/rnz059 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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