On the real-rootedness of the Eulerian transformation

The Eulerian transformation is the linear operator on polynomials in one variable with real coefficients that maps the powers of this variable to the corresponding Eulerian polynomials. The derangement transformation is defined similarly. Brändén and Jochemko have conjectured that the Eulerian transforms of a class of polynomials with nonnegative coefficients, which includes those having all their roots in the interval $[-1,0]$, have only real zeros. This conjecture is proven in this paper. More general transformations are introduced in the combinatorial-geometric context of uniform triangulations of simplicial complexes, where Eulerian and derangement transformations arise in the special case of barycentric subdivision, and are shown to have strong unimodality and gamma-positivity properties. General real-rootedness conjectures for these transformations, which unify various results and conjectures in the literature, are also proposed.