  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.
