Zero Counting and Invariant Sets of Differential Equations

Consider a polynomial vector field ϵ in C ⁿ with algebraic coefficients, and K a compact piece of a trajectory. Let N(K,d) denote the maximal number of isolated intersections between K and an algebraic hypersurface of degree d. We introduce a condition on ϵ called constructible orbits and show that under this condition N(K,d) grows polynomially with d. We establish the constructible orbits condition for linear differential equations over C(t), for planar polynomial differential equations and for some differential equations related to the automorphic j-function. As an application of the main result, we prove a polylogarithmic upper bound for the number of rational points of a given height in planar projections of K following works of Bombieri Pila and Masser.