2016/08/08 by Comte, Georges, Yomdin, Yosef
#14P05 11G50 14G05 30B10 30D15 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1608.02455
For an analytic function f(z)=∑k=0^∞ akzk on a neighbourhood of a closed disc D⊂ \bf C, we give assumptions, in terms of the Taylor coefficients ak of f, under which the number of intersection points of the graph Γf 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, Γf has less than βlogαT rational points of height at most T, for some α, β>0.