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

Abstract : Every polynomial of the form $P = (x + 1)(x^{n−1} + c_1x^{n−2} + \cdots + c_{n−1})$ is representable 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. We give necessary and sufficient conditions upon the possible values of the 8-vector whose components are the number of positive, zero, negative and complex roots of a real polynomial $P$ and the number of positive, zero, negative and complex among the quantities $a_i$ corresponding to $P$. A similar result is proved about entire functions of the form $e^xR$, where $R$ is a polynomial.
Document type :
Journal articles

https://hal.archives-ouvertes.fr/hal-00866116
Submitted on : Thursday, September 26, 2013 - 9:05:05 AM
Last modification on : Monday, October 12, 2020 - 2:28:05 PM

### Identifiers

• HAL Id : hal-00866116, version 1

### Citation

Vladimir Kostov, Vladimir Kostov. A refined realization theorem in the context of the Schur-Szeg\H{o} composition. Bulletin des Sciences Mathématiques, Elsevier, 2012, 136 (5), pp.507-520. ⟨hal-00866116⟩

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