A polynomial vector field on a complex plane C^2 defines a foliation of the plane and one would like to know when the leaves of this foliation are algebraic, i.e. when the analytic integral curves are algebraic. This is a hard unsolved problem. I will suggest a conjectural approach to this problem which uses the reduction of the vector field modulo primes p. It turns out that the corresponding problem in characteristic p is easy to solve. We then conjecture that the original vector field is algebraic if and only if its reduction modulo a prime p is algebraic for almost all p. We have some partial results towards proving this conjecture. This is a work in progress with D. Leshchiner.
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