Local h-polynomials, uniform triangulations and real-rootedness

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The local h-polynomial is a fundamental enumerative invariant
of a triangulation Δ of a simplex. This talk aims to review its
remarkable combinatorial properties, discuss nice examples and address
the following question: Which triangulations of the simplex have a
real-rooted h-polynomial? Our main result states that this is the case
when Δ is the barycentric subdivision of any triangulation of the
simplex. The proof is based on a new combinatorial formula for the
local h-polynomial, which is valid when Δ is any uniform triangulation
of any triangulation of the simplex, and on the study of a family of
polynomials which refine and interpolate between Eulerian and
derangement polynomials. A combinatorial interpretation of the local
h-polynomial of the second barycentric subdivision of the simplex will
be deduced.