Symmetric decompositions, triangulations and real-rootedness

TL;DR

This paper generalizes conditions for polynomials to have nonnegative, real-rooted symmetric decompositions, extending previous results to uniform triangulations and exploring their interlacing properties with applications in geometric combinatorics.

Contribution

It provides a new proof and generalization of symmetric decomposition conditions to subdivision operators in uniform triangulations of simplicial complexes.

Findings

Conditions for interlacing symmetric decompositions are established.

New classes of polynomials with nonnegative, real-rooted symmetric decompositions are identified.

Applications to $f$-vector theory in geometric combinatorics are discussed.

Abstract

Polynomials which afford nonnegative, real-rooted symmetric decompositions have been investigated recently in algebraic, enumerative and geometric combinatorics. Br\"and\'en and Solus have given sufficient conditions under which the image of a polynomial under a certain operator associated to barycentric subdivision has such a decomposition. This paper gives a new proof of their result which generalizes to subdivision operators in the setting of uniform triangulations of simplicial complexes, introduced by the first named author. Sufficient conditions under which these decompositions are also interlacing are described. Applications yield new classes of polynomials in geometric combinatorics which afford nonnegative, real-rooted symmetric decompositions. Some interesting questions in $f$-vector theory arise from this work.