We consider a polynomial vector field with algebraic coefficients, and a compact piece of a trajectory (or separatrix) $K$. Denote by $N(K,d)$ the maximal number of isolated intersections between $K$ and an algebraic hypersurface of degree $d$. We introduce a condition on the vector field called "constructible orbits" and show that under this condition $N(K,d)$ grows polynomially with $d$. We show that this condition holds for every system of linear equations over $C(t)$, for every planar vector field, and for some equations related to modular functions. We also use the main result to produce a polylogarithmic estimate for the number of rational points of a given height in $K$.
I will sketch the ideas behind some of the main statements and mention some possible directions for generalization.