I will explain a joint work with Y. Yomdin (arXiv:1608.02455). For an analytic function $f(z)=\sum^\infty_{k=0}a_kz^k$ on a neighbourhood of a closed disc $D\subset \mathbb{C}$, 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 $\mathbb{Q}[z]$. 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 $\alpha,\beta>0$.