Point counting for foliations over number fields

We consider an algebraic variety and its foliation, both defined over a number field. prove upper bounds for the geometric complexity of intersection between leaf foliation subvariety complementary dimension (also field). Our depend polynomially on degrees, logarithmic heights, distance to certain \emph{locus unlikely intersections}. Under suitable conditions we show that this implies bound, polynomial in degree height, points transcendental sets using such foliations. deduce several results Diophantine geometry. i) Following Masser-Zannier, given pair sections $P,Q$ non-isotrivial family squares elliptic curves do not satisfy constant relation, whenever are simultaneously torsion their order is bounded effectively by degrees log-heights $P,Q$. In particular set simultaneous computable time. ii) Pila, $V\subset\mathbb{C}^n$ there (ineffective) log-height V, discriminants maximal special subvarieties. it follows Andr\'e-Oort powers modular curve decidable time (by algorithm depending universal, ineffective Siegel constant). iii) Schmidt, our counting result Galois-orbit lower bound type previously obtained transcendence methods David.