  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 $E$-polynomials of series-parallel posets and
column-strict labelled Ferrers posets.
