Geodesic
A Realization Theorem in the Context of the Schur–Szegő Composition
Funkcionalʹnyj analiz i ego priloženiâ, Tome 43 (2009) no. 2, pp. 79-83 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $a_i$.
Mots-clés : polynomial
Keywords: Schur–Szegő composition. 
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V. P. Kostov. A Realization Theorem in the Context of the Schur–Szegő Composition. Funkcionalʹnyj analiz i ego priloženiâ, Tome 43 (2009) no. 2, pp. 79-83. http://geodesic.mathdoc.fr/item/FAA_2009_43_2_a7/

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