Zeroes and rational points of analytic functions

For an analytic function $f(z)={\sum }_{k=0}^{\mathrm{\infty }}{a}_k{z}^k$ on a neighbourhood of a closed disc $D\subset \mathbf{C}$, we give assumptions, in terms of the Taylor coefficients $a_k$ of $f$, under which the number of intersection points of the graph ${\mathrm{\Gamma }}_f$ of $f_{\mathrm{\mid }D}$ 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 $\mathbf{Q}[z]$. As a consequence, for any function $f$ in these families, ${\mathrm{\Gamma }}_f$ has less than $\beta {\log }^{\alpha }T$ rational points of height at most $T$, for some $\alpha ,\beta >0$.