We prove that there are absolute constants and such that for every
there are
such that
has at least distinct sign changes in . This improves and extends earlier results of Bloch and Pólya.
Nous prouvons qu’il existe des constantes absolues et telles que pour tout
il existe
tels que
a au moins changements de signe distincts dans . Cela améliore et étend des résultats antérieurs de Bloch et Pólya.
@article{JTNB_2008__20_2_281_0, author = {Tam\'as Erd\'elyi}, 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 = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.627/} }
TY - JOUR 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 DA - 2008/// SP - 281 EP - 287 VL - 20 IS - 2 PB - Université Bordeaux 1 UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.627/ UR - https://zbmath.org/?q=an%3A1163.11022 UR - https://www.ams.org/mathscinet-getitem?mr=2477504 UR - https://doi.org/10.5802/jtnb.627 DO - 10.5802/jtnb.627 LA - en ID - JTNB_2008__20_2_281_0 ER -
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. https://jtnb.centre-mersenne.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: 0713.41004
[2] A. Bloch and G. Pólya, On the roots of certain algebraic equations. Proc. London Math. Soc. 33 (1932), 102–114. | Zbl: 0003.10501
[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. | MR: 1018369 | Zbl: 0629.10024
[4] P. Borwein and T. Erdélyi, Polynomials and polynomial inequalities. Springer-Verlag, New York, 1995. | MR: 1367960 | Zbl: 0840.26002
[5] P. Borwein and T. Erdélyi, On the zeros of polynomials with restricted coefficients. Illinois J. Math. 41 (1997), 667–675. | MR: 1468873 | Zbl: 0906.30005
[6] P. Borwein, T. Erdélyi, and G. Kós, Littlewood-type problems on . Proc. London Math. Soc. 79 (1999), 22–46. | MR: 1687555 | Zbl: 1039.11046
[7] D. Boyd, On a problem of Byrne’s concerning polynomials with restricted coefficients. Math. Comput. 66 (1997), 1697–1703. | Zbl: 0893.12004
[8] P. Erdős and P. Turán, On the distribution of roots of polynomials. Ann. Math. 57 (1950), 105–119. | MR: 33372 | Zbl: 0036.01501
[9] L. K. Hua, Introduction to number theory. Springer-Verlag, Berlin, Heidelberg, New York, 1982. | MR: 665428 | Zbl: 0483.10001
[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: 30713 | Zbl: 0033.14702
[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: 0021.03702
[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. | Zbl: 0007.00101
[14] B. Solomyak, On the random series (an Erdős problem). Ann. Math. 142 (1995), 611–625. | MR: 1356783 | Zbl: 0837.28007
[15] G. Szegő, Bemerkungen zu einem Satz von E. Schmidtüber algebraische Gleichungen. Sitz. Preuss. Akad. Wiss., Phys.-Math. Kl. (1934), 86–98. | Zbl: 0008.38606
Cited by Sources: