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.

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.

Nous prouvons qu’il existe des constantes absolues ${c}_{1}>0$ et ${c}_{2}>0$ telles que pour tout

 $\left\{{a}_{0},{a}_{1},...,{a}_{n}\right\}\subset \left[1,M\right]\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{2em}{0ex}}1\le M\le exp\left({c}_{1}{n}^{1/4}\right)\phantom{\rule{0.166667em}{0ex}},$

il existe

 ${b}_{0},{b}_{1},...,{b}_{n}\in \left\{-1,0,1\right\}$

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

DOI: 10.5802/jtnb.627
Erdélyi, Tamás 1

1 Department of Mathematics Texas A&M University College Station, Texas 77843
@article{JTNB_2008__20_2_281_0,
author = {Erd\'elyi, Tam\'as},
title = {Extensions of the {Bloch{\textendash}P\'olya} theorem on the number of real zeros of polynomials},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {281--287},
publisher = {Universit\'e Bordeaux 1},
volume = {20},
number = {2},
year = {2008},
doi = {10.5802/jtnb.627},
zbl = {1163.11022},
mrnumber = {2477504},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/jtnb.627/}
}
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
SP  - 281
EP  - 287
VL  - 20
IS  - 2
PB  - Université Bordeaux 1
UR  - http://archive.numdam.org/articles/10.5802/jtnb.627/
DO  - 10.5802/jtnb.627
LA  - en
ID  - JTNB_2008__20_2_281_0
ER  - 
%0 Journal Article
%A Erdélyi, Tamás
%T Extensions of the Bloch–Pólya theorem on the number of real zeros of polynomials
%J Journal de théorie des nombres de Bordeaux
%D 2008
%P 281-287
%V 20
%N 2
%I Université Bordeaux 1
%U http://archive.numdam.org/articles/10.5802/jtnb.627/
%R 10.5802/jtnb.627
%G en
%F JTNB_2008__20_2_281_0
Erdélyi, Tamás. 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. http://archive.numdam.org/articles/10.5802/jtnb.627/

[1] F. Amoroso, Sur le diamètre transfini entier d’un intervalle réel. Ann. Inst. Fourier, Grenoble 40 (1990), 885–911. | Numdam | Zbl

[2] A. Bloch and G. Pólya, On the roots of certain algebraic equations. Proc. London Math. Soc. 33 (1932), 102–114. | Zbl

[3] E. Bombieri and J. Vaaler, Polynomials with low height and prescribed vanishing in analytic number theory and Diophantine problems. Birkhäuser, 1987, pp. 53–73. | MR | Zbl

[4] P. Borwein and T. Erdélyi, Polynomials and polynomial inequalities. Springer-Verlag, New York, 1995. | MR | Zbl

[5] P. Borwein and T. Erdélyi, On the zeros of polynomials with restricted coefficients. Illinois J. Math. 41 (1997), 667–675. | MR | Zbl

[6] P. Borwein, T. Erdélyi, and G. Kós, Littlewood-type problems on $\left[0,1\right]$. Proc. London Math. Soc. 79 (1999), 22–46. | MR | Zbl

[7] D. Boyd, On a problem of Byrne’s concerning polynomials with restricted coefficients. Math. Comput. 66 (1997), 1697–1703. | Zbl

[8] P. Erdős and P. Turán, On the distribution of roots of polynomials. Ann. Math. 57 (1950), 105–119. | MR | Zbl

[9] L. K. Hua, Introduction to number theory. Springer-Verlag, Berlin, Heidelberg, New York, 1982. | MR | Zbl

[10] M. Kac, On the average number of real roots of a random algebraic equation, II. Proc. London Math. Soc. 50 (1948), 390–408. | MR | Zbl

[11] J. E. Littlewood and A. C. Offord, On the number of real roots of a random algebraic equation, II. Proc. Cam. Phil. Soc. 35 (1939), 133–148. | Zbl

[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. | Zbl

[14] B. Solomyak, On the random series $\sum ±{\lambda }^{n}$ (an Erdős problem). Ann. Math. 142 (1995), 611–625. | MR | Zbl

[15] G. Szegő, Bemerkungen zu einem Satz von E. Schmidtüber algebraische Gleichungen. Sitz. Preuss. Akad. Wiss., Phys.-Math. Kl. (1934), 86–98. | Zbl

Cited by Sources: