  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 $\alpha, \beta >0$.
