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

On the Bombieri-Pila Method Over Function Fields