Zero counting and invariant sets of differential equations

Abstract

Consider a polynomial vector field $\xi$ in $\mathbb{C}^n$ 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 $\xi$ called \emph{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 $\mathbb{C}(t)$, 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.