Geodesic
A Realization Theorem in the Context of the Schur--Szeg\H{o} Composition
Funkcionalʹnyj analiz i ego priloženiâ, Tome 43 (2009) no. 2, pp. 79-83.

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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--Szeg\H{o} Composition. Funkcionalʹnyj analiz i ego priloženiâ, Tome 43 (2009) no. 2, pp. 79-83. http://geodesic.mathdoc.fr/item/FAA_2009_43_2_a7/

[1] V. V. Prasolov, Mnogochleny, MTsNMO, M., 2003

[2] Q. I. Rahman, G. Schmeisser, Analytic Theory of Polynomials, London Math. Soc. Monographs (N.S.), 26, Oxford Univ. Press, New York, NY, 2002 | MR | Zbl

[3] S. Alkhatib, V. P. Kostov, Revista Mat. Complut., 21:1 (2008), 191–206 | MR | Zbl

[4] V. P. Kostov, C. R. Acad. Sci. Paris Sér. I, 345:9 (2007), 483–488 | DOI | MR | Zbl

[5] V. P. Kostov, B. Z. Shapiro, C. R. Acad. Sci. Paris Sér. I, 343:2 (2006), 81–86 | DOI | MR | Zbl