On the Bombieri-Pila Method Over Function Fields

In 1989 E. Bombieri and J. Pila proved that if G is a subset of an
irreducible algebraic curve of degree d inside a square of side N , then
the number of lattice points on G is bounded
 by c(d, epsilon)N^{1/d +epsilon} for any epsilon > 0, where the
constant c(d, epsilon) does not depend on G.
We will establish a function field analogue of this result.