On the Bombieri-Pila method over function fields

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