Symmetric decompositions, triangulations and real-rootedness

Abstract: A triangulation of a simplicial complex $Δ$ is said to be uniform if the $f$-vector of its restriction to a face of $Δ$ depends only on the dimension of that face. The notion of uniform triangulation was introduced by Christos Athanasiadis in order to conveniently unify many well known types of triangulations such as barycentric, $r$-colored barycentric, $r$-fold edgewise etc. These triangulations have the common feature that, for certain "nice" classes of simplicial complexes $Δ$, the $h$-polynomial of the triangulation $Δ^′$ of $Δ$, is real rooted with nonnegative coefficients. Athanasiadis proved that, uniform triangulations having the so called stong interlacing property, have real rooted $h$-polynomials with nonnegative coefficients.

We continue this line of research and we study under which conditions the $h$-polynomial of a uniform triangulation $Δ^′$ of $Δ$ has a nonnegative real rooted symmetric decomposition. We also provide conditions under which this decomposition is also interlacing. Applications yield new classes of polynomials in geometric combinatorics which afford nonnegative, real-rooted symmetric decompositions. Some interesting questions in $h$-vector theory arise from this work.

This is joint work with Christos Athanasiadis.

Mathematics

Audience: researchers in the topic

Greek Algebra & Number Theory Seminar