ZEROES AND RATIONAL POINTS OF ANALYTIC FUNCTIONS

For an analytic function f (z) = Sigma(infinity)(k=0) (a)k(zk)on a neighbourhood of a closed disc D subset of 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 Gamma(vertical bar 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 algebraic differential equations with coefficients and initial conditions in Q. 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.