Zeroes and rational points of analytic functions

Comte, Georges; Yomdin, Yosef

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

Pour une function analytique $f\left(z\right)={\sum }_{k=0}^{\infty }{a}_k{z}^k$ 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 ${\Gamma }_f$ 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, ${\Gamma }_f$ possède moins de $\beta {log}^{\alpha }T$ points rationnels de hauteur au plus $T$, pour $\alpha ,\beta >0$.

For an analytic function $f\left(z\right)={\sum }_{k=0}^{\infty }{a}_k{z}^k$ 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 ${\Gamma }_f$ 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 $\mathbb{Q}$. As a consequence, for any function $f$ in these families, ${\Gamma }_f$ has less than $\beta {log}^{\alpha }T$ rational points of height at most $T$, for some $\alpha ,\beta >0$.

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

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