For an analytic function f(z)=∑_k=0^∞ a_kz^k on a neighbourhood of a closed disc D⊂ C, we give assumptions, in terms of the Taylor coefficients a_k of f, under which the number of intersection points of the graph Γ_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 linear differential equations with coefficients in Q[z]. As a

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... sequence, for any function f in these families, Γ_f has less than β^α T rational points of height at most T, for some α, β >0.