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Every polynomial of the form $P=(x+1)(x^{n-1}+c_1x^{n-2}+\cdots +c_{n-1})$ isrepresentable 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. Wegive necessary and sufficient conditions upon the possible values of the$8$-vector whose components are the number of positive, zero, negative andcomplex roots of a real polynomial $P$ and the number of positive, zero,negative and complex among the quantities $a_i$ corresponding to $P$. A similarresult is proved about entire functions of the form $e^xR$, where $R$ is apolynomial.

A refined realization theorem in the context of the Schur-Szeg\H{o} composition