Extensions of the Bloch–Pólya theorem on the number of real zeros of polynomials
Journal de théorie des nombres de Bordeaux, Volume 20 (2008) no. 2, p. 281-287

We prove that there are absolute constants ${c}_{1}>0$ and ${c}_{2}>0$ such that for every

$\left\{{a}_{0},{a}_{1},...,{a}_{n}\right\}\subset \left[1,M\right]\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2em}{0ex}}1\le M\le exp\left({c}_{1}{n}^{1/4}\right)\phantom{\rule{0.166667em}{0ex}},$

there are

${b}_{0},{b}_{1},...,{b}_{n}\in \left\{-1,0,1\right\}$

such that

$P\left(z\right)=\sum _{j=0}^{n}{b}_{j}{a}_{j}{z}^{j}$

has at least ${c}_{2}{n}^{1/4}$ distinct sign changes in $\left(0,1\right)$. This improves and extends earlier results of Bloch and Pólya.

Nous prouvons qu’il existe des constantes absolues ${c}_{1}>0$ et ${c}_{2}>0$ telles que pour tout

$\left\{{a}_{0},{a}_{1},...,{a}_{n}\right\}\subset \left[1,M\right]\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2em}{0ex}}1\le M\le exp\left({c}_{1}{n}^{1/4}\right)\phantom{\rule{0.166667em}{0ex}},$

il existe

${b}_{0},{b}_{1},...,{b}_{n}\in \left\{-1,0,1\right\}$

tels que

$P\left(z\right)=\sum _{j=0}^{n}{b}_{j}{a}_{j}{z}^{j}$

a au moins ${c}_{2}{n}^{1/4}$ changements de signe distincts dans $\right]0,1\left[$. Cela améliore et étend des résultats antérieurs de Bloch et Pólya.

@article{JTNB_2008__20_2_281_0,
author = {Erd\'elyi, Tam\'as},
title = {Extensions of the Bloch--P\'olya theorem on the number of real zeros of polynomials},
journal = {Journal de th\'eorie des nombres de Bordeaux},
publisher = {Universit\'e Bordeaux 1},
volume = {20},
number = {2},
year = {2008},
pages = {281-287},
doi = {10.5802/jtnb.627},
mrnumber = {2477504},
zbl = {1163.11022},
language = {en},
url = {http://www.numdam.org/item/JTNB_2008__20_2_281_0}
}

Erdélyi, Tamás. Extensions of the Bloch–Pólya theorem on the number of real zeros of polynomials. Journal de théorie des nombres de Bordeaux, Volume 20 (2008) no. 2, pp. 281-287. doi : 10.5802/jtnb.627. http://www.numdam.org/item/JTNB_2008__20_2_281_0/

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