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

Every polynomial of the form P=(x+1)(xn−1+c1xn−2+⋯+cn−1) is representable as Schur–Szegő composition of n−1 polynomials of the form (x+1)n−1(x+ai), where the numbers ai 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 ai corresponding to P. A similar result is proved about entire functions of the form exR, where R is a polynomial.