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

Tamás Erdélyi[1]

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

Journal de Théorie des Nombres de Bordeaux (2008)

  • Volume: 20, Issue: 2, page 281-287
  • ISSN: 1246-7405

Abstract

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We prove that there are absolute constants c 1 > 0 and c 2 > 0 such that for every { a 0 , a 1 , ... , a n } ⊂ [ 1 , M ] , 1 ≤ M ≤ exp ( c 1 n 1 / 4 ) , there are b 0 , b 1 , ... , b n ∈ { - 1 , 0 , 1 } such that P ( z ) = ∑ 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.

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. 
  13. I. Schur, Untersuchungen über algebraische Gleichungen. Sitz. Preuss. Akad. Wiss., Phys.-Math. Kl. (1933), 403–428. Zbl0007.00101
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