Symmetric decompositions, triangulations and real-rootedness

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.