This paper establishes the necessary and sufficient conditions for the reality of all the zeros in a polynomial sequence ${\left\{P_i\right\}}_{i=1}^{\infty }$ generated by a three-term recurrence relation $P_i\left(x\right)+Q_1\left(x\right)P_{i-1}\left(x\right)+Q_2\left(x\right)P_{i-2}\left(x\right)=0$ with the standard initial conditions $P_0\left(x\right)=1,P_{-1}\left(x\right)=0,$ where $Q_1\left(x\right)$ and $Q_2\left(x\right)$ are arbitrary real polynomials.

∗ Research partially supported by INTAS grant 97-1644A real polynomial of one real variable is hyperbolic (resp. strictly hyperbolic) if it has only real roots (resp. if its roots are real and distinct). We prove that there are 116 possible non-degenerate configurations between the roots of a degree 5 strictly hyperbolic polynomial and of its derivatives (i.e. configurations without equalities between roots). The standard Rolle theorem allows 286 such configurations. To obtain the result we study...

2000 Mathematics Subject Classification: 12D10.In the present paper we consider degree 6 hyperbolic polynomials (HPs) in one variable (i.e. real and with all roots real). We are interested in such HPs whose number of equalities between roots of the polynomial and/or its derivatives is higher than expected. We give the complete study of the four families of such degree 6 even HPs and also of HPs which are primitives of degree 5 HPs.Research partially supported by research project 20682 for cooperation...

We prove that there are absolute constants $c_1\>0$ and $c_2\>0$ such that for every$$\{a_0,a_1,...,a_n\}\subset [1,M],1\le M\le exp\left(c_1{n}^{1/4}\right),$$there are$$b_0,b_1,...,b_n\in \{-1,0,1\}$$such that$$P\left(z\right)=\sum _{j=0}^n{b}_j{a}_j{z}^j$$has at least $c_2{n}^{1/4}$ distinct sign changes in $(0,1)$. This improves and extends earlier results of Bloch and Pólya.