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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  1. F. Amoroso, Sur le diamètre transfini entier d’un intervalle réel. Ann. Inst. Fourier, Grenoble 40 (1990), 885–911. Zbl0713.41004
  2. A. Bloch and G. Pólya, On the roots of certain algebraic equations. Proc. London Math. Soc. 33 (1932), 102–114. Zbl0003.10501
  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. Zbl0629.10024MR1018369
  4. P. Borwein and T. Erdélyi, Polynomials and polynomial inequalities. Springer-Verlag, New York, 1995. Zbl0840.26002MR1367960
  5. P. Borwein and T. Erdélyi, On the zeros of polynomials with restricted coefficients. Illinois J. Math. 41 (1997), 667–675. Zbl0906.30005MR1468873
  6. P. Borwein, T. Erdélyi, and G. Kós, Littlewood-type problems on [ 0 , 1 ] . Proc. London Math. Soc. 79 (1999), 22–46. Zbl1039.11046MR1687555
  7. D. Boyd, On a problem of Byrne’s concerning polynomials with restricted coefficients. Math. Comput. 66 (1997), 1697–1703. Zbl0893.12004
  8. P. Erdős and P. Turán, On the distribution of roots of polynomials. Ann. Math. 57 (1950), 105–119. Zbl0036.01501MR33372
  9. L. K. Hua, Introduction to number theory. Springer-Verlag, Berlin, Heidelberg, New York, 1982. Zbl0483.10001MR665428
  10. M. Kac, On the average number of real roots of a random algebraic equation, II. Proc. London Math. Soc. 50 (1948), 390–408. Zbl0033.14702MR30713
  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. Zbl0021.03702
  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
  14. B. Solomyak, On the random series ∑ ± λ n (an Erdős problem). Ann. Math. 142 (1995), 611–625. Zbl0837.28007MR1356783
  15. G. Szegő, Bemerkungen zu einem Satz von E. Schmidtüber algebraische Gleichungen. Sitz. Preuss. Akad. Wiss., Phys.-Math. Kl. (1934), 86–98. Zbl0008.38606

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