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Article Dans Une Revue Annales de l'Institut Fourier Année : 2018

Zeroes and rational points of analytic functions

Résumé

For an analytic function $f(z)=\sum_{k=0}^\infty a_kz^k$ on a neighbourhood of a closed disc $D\subset {\bf C}$, 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 $f_{\vert D}$ 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 linear differential equations with coefficients in ${\bf Q}[z]$. 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 $\alpha, \beta >0$.
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Dates et versions

hal-03798604 , version 1 (05-10-2022)

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Georges Comte, Yosef Yomdin. Zeroes and rational points of analytic functions. Annales de l'Institut Fourier, 2018. ⟨hal-03798604⟩
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