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Interlacing properties and the Schur-Szegő composition. (English) Zbl 1245.12002

Summary: Each degree \(n\) polynomial in one variable of the form \((x+1)(x^{n-1}+c_{1} x^{n-2}+\cdots +c_{n-1})\) is representable in a unique way as a Schur-Szegő composition of \(n-1\) polynomials of the form \((x+1)^{n-1} (x+a_i)\), see the author [C. R., Math., Acad. Sci. Paris 345, No. 9, 483–488 (2007; Zbl 1164.12001)], the author and S. Alkhatib [Rev. Mat. Complut. 21, No. 1, 191–206 (2008; Zbl 1166.12001)] and the author [Math. Balk., New Ser. 22, No. 1-2, 155–173 (2008; Zbl 1166.12002)]. Set \(\sigma_{j} := \sum_{1\leq i_{1}<\cdots <i_{j}\leq n-1}a_{i_{1}}\cdots a_{i_{j}}\). The eigenvalues of the affine mapping \((c_{1},\dots ,c_{n-1})\mapsto (\sigma_{1},\dots ,\sigma_{n-1})\) are positive rational numbers and its eigenvectors are defined by hyperbolic polynomials (i.e. with real roots only). In the present paper we prove interlacing properties of the roots of these polynomials.

MSC:

12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
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References:

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