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Location of zeros of families of trinomials. (Localisation de zéros de familles de trinômes.) (French) Zbl 0962.30002
Let \(n\) be a natural number \(\varphi= (1+\sqrt 5)/2\), and let \(a\) be a real number. In this paper the authors obtain explicit estimations, by using three different methods, of some roots of the following families of trinomials: \[ f_{n,a}(z)= z^{n+ 1}- az+ a-1, \] \[ G_n(z)= (\varphi- 1) z^{n+1}- \varphi z^n+ 1 \] and \[ H_n(z)= \varphi z^{n+ 1}-(\varphi- 1) z^n- 1. \] The first method is based on finding the approaching solutions of the equation. The second method is given by using the implicit function theorem, and the third method provides an expansion in hypergeometric series of solutions of a trinomial.
MSC:
30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)
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