# ANNALES DE L'INSTITUT FOURIER

Racines de polynômes de Bernstein
Annales de l'Institut Fourier, Tome 36 (1986) no. 4, pp. 1-30.

On considère un polynôme $P$, à coefficients réels non négatifs, à deux indéterminées. On montre que la connaissance des pôles des intégrales

 ${\int }_{0}^{1}{\int }_{0}^{1}{x}_{1}^{{\beta }_{1}-1}{x}_{2}^{{\beta }_{2}-1}P\left({x}_{1},{x}_{2}{\right)}^{s}d{x}_{1}d{x}_{2}$

donne des renseignements sur les racines du polynômes de Bernstein de $P$. La détermination des pôles des intégrales peut se faire en utilisant certaines méthodes de Mellin. Des calculs explicites sont donnés.

Let $P$ be a polynomial with non negative real coefficients, in two indeterminates. One shows that the knowledge of the poles of the integrals

 ${\int }_{0}^{1}{\int }_{0}^{1}{x}_{1}^{{\beta }_{1}-1}{x}_{2}^{{\beta }_{2}-1}P\left({x}_{1},{x}_{2}{\right)}^{s}d{x}_{1}d{x}_{2}$

gives some of the roots of the Bernstein polynomial of $P$. One can calculate poles of these integrals using some Mellin’s methods. Some explicit computations are given.

@article{AIF_1986__36_4_1_0,
author = {Cassou-Nogu\es, Pierrette},
title = {Racines de polyn\^omes de Bernstein},
journal = {Annales de l'Institut Fourier},
pages = {1--30},
publisher = {Imprimerie Durand},
volume = {36},
number = {4},
year = {1986},
doi = {10.5802/aif.1067},
zbl = {0597.32004},
mrnumber = {88c:32012},
language = {fr},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1067/}
}
Cassou-Noguès, Pierrette. Racines de polynômes de Bernstein. Annales de l'Institut Fourier, Tome 36 (1986) no. 4, pp. 1-30. doi : 10.5802/aif.1067. https://aif.centre-mersenne.org/articles/10.5802/aif.1067/`

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