Leonardo Saud Maia Leite: Totally nonnegative matrices, chain enumeration and zeros of polynomials

Abstract: We prove that any lower triangular and totally nonnegative matrix whose diagonal entries are all equal to one gives rise to a family real-rooted polynomials. This is used to develop a general theory for proving that chain polynomials of rank uniform posets are real-rooted. The results obtained extend and unify results of the first author, Brenti, Welker and Athanasiadis. In the process we define a notion of h-vectors for a large class of posets which generalize the h-vectors commonly associated to simplicial and cubical complexes.