Extensions of the Bloch–Pólya theorem on the number of real zeros of polynomials

Erdélyi, Tamás

Journal de théorie des nombres de Bordeaux, Tome 20 (2008) no. 2, pp. 281-287.

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

$$\{a_0,a_1,...,a_n\}\subset [1,M],1\le M\le exp\left(c_1{n}^{1/4}\right),$$

il existe

$$b_0,b_1,...,b_n\in \{-1,0,1\}$$

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 $]0,1[$. Cela améliore et étend des résultats antérieurs de Bloch et Pólya.

We prove that there are absolute constants $c_1>0$ and $c_2>0$ such that for every

$$\{a_0,a_1,...,a_n\}\subset [1,M],1\le M\le exp\left(c_1{n}^{1/4}\right),$$

there are

$$b_0,b_1,...,b_n\in \{-1,0,1\}$$

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 $(0,1)$. This improves and extends earlier results of Bloch and Pólya.

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