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

Tamás Erdélyi

doi:10.5802/jtnb.627

Tamás Erdélyi ¹

¹ Department of Mathematics Texas A&M University College Station, Texas 77843

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

Abstract
Résumé

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 \leq M \leq exp (c_{1} n^{1 / 4}),

there are

b_{0}, b_{1}, ..., b_{n} \in {- 1, 0, 1}

such that

P (z) = \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.

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 \leq M \leq exp (c_{1} n^{1 / 4}),

il existe

b_{0}, b_{1}, ..., b_{n} \in {- 1, 0, 1}

tels que

P (z) = \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.

MR: 2477504 | Zbl: 1163.11022

DOI: 10.5802/jtnb.627

Author's affiliations:

Tamás Erdélyi ¹

¹ Department of Mathematics Texas A&M University College Station, Texas 77843

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     journal = {Journal de th\'eorie des nombres de Bordeaux},
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     publisher = {Universit\'e Bordeaux 1},
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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, Volume 20 (2008) no. 2, pp. 281-287. doi : 10.5802/jtnb.627. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.627/

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