Zeroes and rational points of analytic functions

Comte, Georges; Yomdin, Yosef

doi:10.5802/aif.3213

Zeroes and rational points of analytic functions
[Zéros et points rationnels des fonctions analytiques]

Comte, Georges ¹ ; Yomdin, Yosef ²

¹ Univ. Grenoble Alpes, Univ. Savoie Mont Blanc CNRS, LAMA 73000 Chambéry (France)
² Department of Mathematics The Weizmann Institute of Science Rehovot 76100 (Israel)

Annales de l'Institut Fourier, Tome 68 (2018) no. 6, pp. 2445-2476.

Résumé
Abstract

Pour une function analytique $f (z) = \sum_{k = 0}^{\infty} a_{k} z^{k}$ définie sur un voisinage d’un disque fermé $D \subset ℂ$ , nous donnons des conditions, portant sur les coefficients de Taylor $a_{k}$ de $f$ , sous lesquelles le nombre de points d’intersection du graphe $Γ_{f}$ de $f_{| D}$ avec les courbes algébriques de degré $d$ est polynomialement borné en $d$ . Nous montrons en particulier que ces conditions sont satisfaites pour les séries entières aléatoires, pour certaines classes explicites de séries lacunaires, et pour les solutions d’équations différentielles algébriques avec coefficients et conditions initiales rationnels. En conséquence, pour toute fonction $f$ dans une de ces familles, $Γ_{f}$ possède moins de $β {log}^{α} T$ points rationnels de hauteur au plus $T$ , pour $α, β > 0$ .

For an analytic function $f (z) = \sum_{k = 0}^{\infty} a_{k} z^{k}$ on a neighbourhood of a closed disc $D \subset ℂ$ , we give assumptions, in terms of the Taylor coefficients $a_{k}$ of $f$ , under which the number of intersection points of the graph $Γ_{f}$ of $f_{| D}$ and algebraic curves of degree $d$ is polynomially bounded in $d$ . In particular, we show these assumptions are satisfied for random power series, for some explicit classes of lacunary series, and for solutions of algebraic differential equations with coefficients and initial conditions in $ℚ$ . As a consequence, for any function $f$ in these families, $Γ_{f}$ has less than $β {log}^{α} T$ rational points of height at most $T$ , for some $α, β > 0$ .

Reçu le : 2016-09-23
Révisé le : 2017-11-02
Accepté le : 2017-12-13
Publié le : 2018-11-23

DOI : 10.5802/aif.3213

Classification : 14G05, 30B10, 30B20, 30D15
Keywords: zeroes of analytic functions, rational points
Mot clés : zéros des fonctions analytiques, points rationnels

Affiliations des auteurs :

Comte, Georges ¹ ; Yomdin, Yosef ²

¹ Univ. Grenoble Alpes, Univ. Savoie Mont Blanc CNRS, LAMA 73000 Chambéry (France)
² Department of Mathematics The Weizmann Institute of Science Rehovot 76100 (Israel)

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{AIF_2018__68_6_2445_0,
     author = {Comte, Georges and Yomdin, Yosef},
     title = {Zeroes and rational points of analytic functions},
     journal = {Annales de l'Institut Fourier},
     pages = {2445--2476},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {68},
     number = {6},
     year = {2018},
     doi = {10.5802/aif.3213},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3213/}
}

TY  - JOUR
AU  - Comte, Georges
AU  - Yomdin, Yosef
TI  - Zeroes and rational points of analytic functions
JO  - Annales de l'Institut Fourier
PY  - 2018
SP  - 2445
EP  - 2476
VL  - 68
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3213/
DO  - 10.5802/aif.3213
LA  - en
ID  - AIF_2018__68_6_2445_0
ER  -

%0 Journal Article
%A Comte, Georges
%A Yomdin, Yosef
%T Zeroes and rational points of analytic functions
%J Annales de l'Institut Fourier
%D 2018
%P 2445-2476
%V 68
%N 6
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3213/
%R 10.5802/aif.3213
%G en
%F AIF_2018__68_6_2445_0

Comte, Georges; Yomdin, Yosef. Zeroes and rational points of analytic functions. Annales de l'Institut Fourier, Tome 68 (2018) no. 6, pp. 2445-2476. doi : 10.5802/aif.3213. https://aif.centre-mersenne.org/articles/10.5802/aif.3213/

Bibliographie
Cité par

[1] Baker, George A. Jr.; Lubinsky, Doron S. Convergence theorems for rows of differential and algebraic Hermite-Padé approximations, J. Comput. Appl. Math., Volume 18 (1987) no. 1, pp. 29-52 Special issue on the 11th South African symposium on numerical mathematics (Umhlanga, 1985) | DOI | MR | Zbl

[2] Bautin, N. N. Du nombre de cycles limites naissant en cas de variation des coefficients d’un état d’équilibre du type foyer ou centre, C. R. (Dokl.) Acad. Sci. URSS (N. S.), Volume 24 (1939), pp. 669-672 | MR | Zbl

[3] Bautin, N. N. On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Am. Math. Soc., Transl., Volume 1954 (1954) no. 100, 19 pages (19 p.) | MR | Zbl

[4] Besson, Étienne Sur le nombre de points rationnels de la fonction zêta de Riemann, 2011 Mémoire de Master 2. École Normale Supérieure de Lyon (France)

[5] Besson, Étienne Points rationnels de la fonction gamma d’Euler, Arch. Math., Volume 103 (2014) no. 1, pp. 61-73 | DOI | MR | Zbl

[6] Binyamini, Gal Multiplicity estimates: a Morse-theoretic approach, Duke Math. J., Volume 165 (2016) no. 1, pp. 95-128 | DOI | MR | Zbl

[7] Binyamini, Gal Zero counting and invariant sets of differential equations (2016) (to appear in Int. Math. Res. Not., https://arxiv.org/abs/1510.00120)

[8] Binyamini, Gal; Novikov, Dmitry The Pila-Wilkie theorem for subanalytic families: a complex analytic approach, Compos. Math., Volume 153 (2016) no. 10, pp. 2171-2194 | Zbl

[9] Binyamini, Gal; Novikov, Dmitry Wilkie’s conjecture for restricted elementary functions, Ann. Math., Volume 186 (2017) no. 1, pp. 237-275 | DOI | MR | Zbl

[10] Bombieri, Enrico; Pila, Jonathan The number of integral points on arcs and ovals, Duke Math. J., Volume 59 (1989) no. 2, pp. 337-357 | DOI | MR | Zbl

[11] Bos, Len P.; Brudnyĭ, Alexander; Levenberg, Norman On polynomial inequalities on exponential curves in $ℂ^{n}$ , Constr. Approx., Volume 31 (2010) no. 1, pp. 139-147 | DOI | MR | Zbl

[12] Boxall, Gareth J.; Jones, Gareth O. Algebraic values of certain analytic functions, Int. Math. Res. Not. (2015) no. 4, pp. 1141-1158 | MR | Zbl

[13] Boxall, Gareth J.; Jones, Gareth O. Rational values of entire functions of finite order, Int. Math. Res. Not. (2015) no. 22, pp. 12251-12264 | MR | Zbl

[14] Brudnyĭ, Alexander On local behavior of analytic functions, J. Funct. Anal., Volume 169 (1999) no. 2, pp. 481-493 | DOI | MR | Zbl

[15] Brudnyĭ, Alexander Cyclicity of zeroes of families of analytic functions, Geom. Funct. Anal., Volume 13 (2003) no. 6, pp. 1161-1188 | Zbl

[16] Brudnyĭ, Alexander On local behavior of holomorphic functions along complex submanifolds of $ℂ^{n}$ , Invent. Math., Volume 173 (2008) no. 2, pp. 315-363 | Zbl

[17] Brudnyĭ, Alexander Rolle type theorem for cyclicity of zeroes of families of analytic functions, Isr. J. Math., Volume 206 (2015) no. 2, pp. 95-107 | Zbl

[18] Brudnyĭ, Yuri. A.; Ganzburg, Michael I. A certain extremal problem for polynomials in $n$ variables, Izv. Akad. Nauk SSSR Ser. Mat., Volume 37 (1973), pp. 344-355 | MR | Zbl

[19] Cluckers, Raf; Pila, Jonathan; Wilkie, Alex J. Uniform parameterization of subanalytic sets and diophantine applications (2016) (https://arxiv.org/abs/1605.05916)

[20] Coman, Dan; Poletsky, Evgeny A. Bernstein-Walsh inequalities and the exponential curve in $ℂ^{2}$ , Proc. Am. Math. Soc., Volume 131 (2003) no. 3, pp. 879-887 | DOI | MR | Zbl

[21] Coman, Dan; Poletsky, Evgeny A. Measures of transcendency for entire functions, Mich. Math. J., Volume 51 (2003) no. 3, pp. 575-591 | DOI | MR | Zbl

[22] Coman, Dan; Poletsky, Evgeny A. Transcendence measures and algebraic growth of entire functions, Invent. Math., Volume 170 (2007) no. 1, pp. 103-145 | DOI | MR | Zbl

[23] Comte, Georges; Miller, Chris Points of bounded height on oscillatory sets (2017) (to appear in Q. J. Math., https://arxiv.org/abs/1601.03033)

[24] Friedland, Omer; Yomdin, Yosef $(s, p)$ -valent functions, Geometric aspects of functional analysis (Lecture Notes in Mathematics), Volume 2169, Springer, 2017, pp. 123-136 | MR | Zbl

[25] Gabrielov, Andrei Multiplicity of a zero of an analytic function on a trajectory of a vector field, The Arnoldfest (Toronto, ON, 1997) (Fields Institute Communications), Volume 24, American Mathematical Society, 1999, pp. 191-200 | MR | Zbl

[26] Gwoździewicz, Janusz; Kurdyka, Krzysztof; Parusiński, Adam On the number of solutions of an algebraic equation on the curve $y = e^{x} + sin x$ , $x > 0$ , and a consequence for o-minimal structures, Proc. Am. Math. Soc., Volume 127 (1999) no. 4, pp. 1057-1064 | DOI | MR | Zbl

[27] Hille, Einar Analytic function theory. Vol. II, Introductions to Higher Mathematics, Ginn and Company, 1962, xii+496 pages | MR | Zbl

[28] Jones, Gareth O.; Miller, Daniel J.; Thomas, Margaret E. M. Mildness and the density of rational points on certain transcendental curves, Notre Dame J. Formal Logic, Volume 52 (2011) no. 1, pp. 67-74 | DOI | MR | Zbl

[29] Jones, Gareth O.; Thomas, Margaret E. M. The density of algebraic points on certain Pfaffian surfaces, Q. J. Math., Volume 63 (2012) no. 3, pp. 637-651 | DOI | MR | Zbl

[30] Masser, David Rational values of the Riemann zeta function, J. Number Theory, Volume 131 (2011) no. 11, pp. 2037-2046 | DOI | MR | Zbl

[31] Nesterenko Estimates for the number of zeros of certain functions, New advances in transcendence theory (Durham, 1986), Cambridge University Press, 1988, pp. 263-269 | MR | Zbl

[32] Nesterenko Algebraic independence, Narosa Publishing House, 2009, viii+162 pages | MR | Zbl

[33] Nikishin, Evgenii M.; Sorokin, Vladimir N. Rational approximations and orthogonality, Translations of Mathematical Monographs, 92, American Mathematical Society, 1991, viii+221 pages (Translated from the Russian by Ralph P. Boas) | MR | Zbl

[34] Pila, Jonathan Integer points on the dilation of a subanalytic surface, Q. J. Math., Volume 55 (2004) no. 2, pp. 207-223 | DOI | MR | Zbl

[35] Pila, Jonathan Mild parameterization and the rational points of a Pfaff curve, Comment. Math. Univ. St. Pauli, Volume 55 (2006) no. 1, pp. 1-8 | MR | Zbl

[36] Pila, Jonathan; Wilkie, Alex J. The rational points of a definable set, Duke Math. J., Volume 133 (2006) no. 3, pp. 591-616 | DOI | MR | Zbl

[37] Roytwarf, N.; Yomdin, Yosef Bernstein classes, Ann. Inst. Fourier, Volume 47 (1997) no. 3, pp. 825-858 | MR | Zbl

[38] Sadullaev, Azimbai Plurisubharmonic measures and capacities on complex manifolds, Usp. Mat. Nauk, Volume 36 (1981) no. 4, pp. 53-105 | MR | Zbl

[39] Sadullaev, Azimbai Estimates of polynomials on analytic sets, Izv. Akad. Nauk SSSR Ser. Mat., Volume 46 (1982) no. 3, pp. 524-534 | MR | Zbl

[40] Shidlovskii, Andrei Borisovich Transcendental numbers, de Gruyter Studies in Mathematics, 12, Walter de Gruyter, 1989, xx+466 pages (Translated from the Russian by Neal Koblitz, With a foreword by W. Dale Brownawell) | DOI | MR | Zbl

[41] Surroca, Andrea Sur le nombre de points algébriques où une fonction analytique transcendante prend des valeurs algébriques, C. R. Math. Acad. Sci. Paris, Volume 334 (2002) no. 9, pp. 721-725 | DOI | MR | Zbl

[42] Surroca, Andrea Valeurs algébriques de fonctions transcendantes, Int. Math. Res. Not. (2006), 16834, 31 pages (Art. ID 16834, 31 p.) | DOI | MR | Zbl

[43] Tijdeman, Robert On the number of zeros of general exponential polynomials, Nederl. Akad. Wet. Proc. Ser. A, Volume 74 (1971), pp. 1-7 | MR | Zbl

[44] Yomdin, Yosef Oscillation of analytic curves, Proc. Am. Math. Soc., Volume 126 (1998) no. 2, pp. 357-364 | DOI | MR | Zbl

[45] Yomdin, Yosef Global finitness properties of analytic families and algebra of their coefficients, The Arnoldfest (Fields Institute Communications), Volume 24, American Mathematical Society, 1999, pp. 527-555 | Zbl

Cité par Sources :

ANNALES DE L'INSTITUT FOURIER

ARTICLES À PARAÎTRE

FEUILLETER

PRESENTATION

COMITÉ DE RÉDACTION

SOUMETTRE UN ARTICLE

ABONNEMENT