A Realization Theorem in the Context of the Schur–Szegő Composition

Every real polynomial of degree $n$ in one variable with root $-1$ can be represented as the Schur–Szegő composition of $n-1$ polynomials of the form $(x+1)^{n-1}(x+a_i)$, where the numbers $a_i$ are uniquely determined up to permutation. Some $a_i$ are real, and the others form complex conjugate pairs. In this note, we show that for each pair $(\rho,r)$, where $0\le \rho,r\le [n/2]$, there exists a polynomial with exactly $\rho$ pairs of complex conjugate roots and exactly $r$ complex conjugate pairs in the corresponding set of numbers $a_i$.