We prove Pólya’s conjecture of 1943: For a real entire function of order greater than 2 with finitely many non-real zeros, the number of non-real zeros of the nth derivative tends to infinity, as $n\to\infty$. We use the saddle point method and potential theory, combined with the theory of analytic functions with positive imaginary part in the upper half-plane.

Let χ be an analytic vector field defined in a real-analytic manifold of dimension three. We prove that all the singularities of χ can be made elementary by a finite number of blowing-ups in the ambient space.