A mapping connected with the Schur-Szegő composition (Q1046534)

For a couple of real or complex polynomials \(P(x)=\sum_{j=0}^{n}p_{j}x^{j}\) and \(Q(x)=\sum_{j=0}^{n}q_{j}x^{j}\), their Schur-Szegő composition is defined by \((P {* \atop n}Q)(x)=\sum_{j=0}^{n}(p_{j}q_{j}/C_{n}^{j})x^{j}\). Every monic polynomial in one variable of the form \((x+1)S\), \(\deg\,S=n-1\), is presentable in a unique way as a Schur-Szegő composition of \(n-1\) polynomials of the form \((x+1)^{n-1}(x+a_{i})\). In this paper, the author proves geometric properties of the affine mapping associating to the coefficients of \(S\) the \((n-1)\)-tuple of values of the elementary symmetric functions of the numbers \(a_{i}\).