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

Tamás Erdélyi

doi:10.5802/jtnb.627

Tamás Erdélyi ¹

¹Department 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 (Original language)
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.

Article information
Cite this article

Received: 2008-01-28
Published online: 2009-02-24

Zbl MR

DOI: 10.5802/jtnb.627

Author's affiliations:

Tamás Erdélyi ¹

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

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

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