A refined realization theorem in the context of the Schur-Szeg\H{o} composition
Résumé
Every polynomial of the form $P = (x + 1)(x^{n−1} + c_1x^{n−2} + \cdots + c_{n−1})$ is representable as Schur-Szeg\H{o} composition of $n−1$ polynomials of the form $(x +1)^{n−1}(x +a_i )$, where the numbers $a_i$ are unique up to permutation. We give necessary and sufficient conditions upon the possible values of the 8-vector whose components are the number of positive, zero, negative and complex roots of a real polynomial $P$ and the number of positive, zero, negative and complex among the quantities $a_i$ corresponding to $P$. A similar result is proved about entire functions of the form $e^xR$, where $R$ is a polynomial.