5 research outputs found
On operators on polynomials preserving real-rootedness and the Neggers-Stanley Conjecture
We refine a technique used in a paper by Schur on real-rooted polynomials.
This amounts to an extension of a theorem of Wagner on Hadamard products of
Toeplitz matrices. We also apply our results to polynomials for which the
Neggers-Stanley Conjecture is known to hold. More precisely, we settle
interlacing properties for -polynomials of series-parallel posets and
column-strict labelled Ferrers posets
A unified approach to polynomial sequences with only real zeros
We give new sufficient conditions for a sequence of polynomials to have only
real zeros based on the method of interlacing zeros. As applications we derive
several well-known facts, including the reality of zeros of orthogonal
polynomials, matching polynomials, Narayana polynomials and Eulerian
polynomials. We also settle certain conjectures of Stahl on genus polynomials
by proving them for certain classes of graphs, while showing that they are
false in general.Comment: 19 pages, Advances in Applied Mathematics, in pres