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Tamás Erdélyi
Extensions of the Bloch–Pólya theorem on the number of real zeros of polynomials
Journal de théorie des nombres de Bordeaux, 20 no. 2 (2008), p. 281-287, doi: 10.5802/jtnb.627
Article PDF | Analyses MR 2477504 | Zbl 1163.11022

Résumé - Abstract

Nous prouvons qu’il existe des constantes absolues $c_1 > 0$ et $c_2 > 0$ telles que pour tout

$$\lbrace a_0,a_1, \ldots , a_n\rbrace \subset [1,M]\,, \qquad 1 \le M \le \exp (c_1n^{1/4})\,,$$

il existe

$$b_0,b_1,\ldots , b_n \in \lbrace -1,0,1\rbrace $$

tels que

$$P(z) = \sum _{j=0}^n{b_ja_jz^j}$$

a au moins $c_2n^{1/4}$ changements de signe distincts dans $]0,1[$. Cela améliore et étend des résultats antérieurs de Bloch et Pólya.

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