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.