  E. Bombieri and J. Pila introduced a method for bounding the number of
integral lattice points that belong to a given arc under several assumptions.
In this paper we generalize the Bombieri-Pila method to the case of function
fields of genus 0 in one variable. We then apply the result to counting the
number of elliptic curves contain in an isomorphism class and with coefficients
in a box.
