A refined realization theorem in the context of the Schur–Szegő composition

Abstract

Every polynomial of the form P = (x + 1)(x n−1 + c 1 x n−2 + · · · + c n−1 ) is representable as Schur-Szegő 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 x R, where R is a polynomial.2000 Mathematics Subject Classification. Primary 12A10 Secondary 30D99.