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

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{FAA_2009_43_2_a7,
     author = {V. P. Kostov},
     title = {A {Realization} {Theorem} in the {Context} of the {Schur--Szeg\H{o}} {Composition}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {79--83},
     publisher = {mathdoc},
     volume = {43},
     number = {2},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2009_43_2_a7/}
}
TY  - JOUR
AU  - V. P. Kostov
TI  - A Realization Theorem in the Context of the Schur--Szeg\H{o} Composition
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2009
SP  - 79
EP  - 83
VL  - 43
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2009_43_2_a7/
LA  - ru
ID  - FAA_2009_43_2_a7
ER  - 
%0 Journal Article
%A V. P. Kostov
%T A Realization Theorem in the Context of the Schur--Szeg\H{o} Composition
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2009
%P 79-83
%V 43
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2009_43_2_a7/
%G ru
%F FAA_2009_43_2_a7
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/