On arrangements of real roots of a real polynomial and its derivatives

We prove that all arrangements (consistent with the Rolle theorem and some other natural restrictions) of the real roots of a real polynomial and of its $s$-th derivative are realizable by real polynomials.Comment: To appear in Serdica Math. J. 29 (2003

Polynominals related to powers of the Dedekind eta function

The vanishing properties of Fourier coefficients of integral powers of the Dedekind eta function correspond to the existence of integral roots of integer-valued polynomials Pn(x) introduced by M. Newman. In this paper we study the derivatives of these polynomials. We obtain non-vanishing results at integral points. As an application we prove that integral roots are simple if the index n of the polynomial is equal to a prime power pm or to pm + 1. We obtain a formula for the derivative of Pn(x) involving the polynomials of lower degree