For an analytic function on a neighbourhood of a closed disc , we give assumptions, in terms of the Taylor coefficients of , under which the number of intersection points of the graph of and algebraic curves of degree is polynomially bounded in . 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 . As a consequence, for any function in these families, has less than rational points of height at most , for some .
Pour une function analytique définie sur un voisinage d’un disque fermé , nous donnons des conditions, portant sur les coefficients de Taylor de , sous lesquelles le nombre de points d’intersection du graphe de avec les courbes algébriques de degré est polynomialement borné en . 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 dans une de ces familles, possède moins de points rationnels de hauteur au plus , pour .
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Keywords: zeroes of analytic functions, rational points
Mot clés : zéros des fonctions analytiques, points rationnels
Comte, Georges 1; Yomdin, Yosef 2
@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, Volume 68 (2018) no. 6, pp. 2445-2476. doi : 10.5802/aif.3213. https://aif.centre-mersenne.org/articles/10.5802/aif.3213/
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