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

• [1] Department of Mathematics Texas A&M University College Station, Texas 77843
• Volume: 20, Issue: 2, page 281-287
• ISSN: 1246-7405

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## Abstract

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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.

## How to cite

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Erdélyi, Tamás. "Extensions of the Bloch–Pólya theorem on the number of real zeros of polynomials." Journal de Théorie des Nombres de Bordeaux 20.2 (2008): 281-287. <http://eudml.org/doc/10836>.

@article{Erdélyi2008,
abstract = {We prove that there are absolute constants $c_1 &gt; 0$ and $c_2 &gt; 0$ such that for every$\lbrace a\_0,a\_1, \ldots , a\_n\rbrace \subset [1,M]\,, \qquad 1 \le M \le \exp (c\_1n^\{1/4\})\,,$there are$b\_0,b\_1,\ldots , b\_n \in \lbrace -1,0,1\rbrace$such that$P(z) = \sum \_\{j=0\}^n\{b\_ja\_jz^j\}$has at least $c_2n^\{1/4\}$ distinct sign changes in $(0,1)$. This improves and extends earlier results of Bloch and Pólya.},
affiliation = {Department of Mathematics Texas A&M University College Station, Texas 77843},
author = {Erdélyi, Tamás},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {2},
pages = {281-287},
publisher = {Université Bordeaux 1},
title = {Extensions of the Bloch–Pólya theorem on the number of real zeros of polynomials},
url = {http://eudml.org/doc/10836},
volume = {20},
year = {2008},
}

TY - JOUR
AU - Erdélyi, Tamás
TI - Extensions of the Bloch–Pólya theorem on the number of real zeros of polynomials
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2008
PB - Université Bordeaux 1
VL - 20
IS - 2
SP - 281
EP - 287
AB - We prove that there are absolute constants $c_1 &gt; 0$ and $c_2 &gt; 0$ such that for every$\lbrace a_0,a_1, \ldots , a_n\rbrace \subset [1,M]\,, \qquad 1 \le M \le \exp (c_1n^{1/4})\,,$there are$b_0,b_1,\ldots , b_n \in \lbrace -1,0,1\rbrace$such that$P(z) = \sum _{j=0}^n{b_ja_jz^j}$has at least $c_2n^{1/4}$ distinct sign changes in $(0,1)$. This improves and extends earlier results of Bloch and Pólya.
LA - eng
UR - http://eudml.org/doc/10836
ER -

## References

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12. E. Schmidt, Über algebraische Gleichungen vom Pólya–Bloch-Typos. Sitz. Preuss. Akad. Wiss., Phys.-Math. Kl. (1932), 321.
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