Explicit Hilbert’s irreducibility theorem in function fields

We prove a quantitative version of Hilbert's irreducibility theorem for function fields: If $f(T_1,\ldots, T_n,X)$ is an irreducible polynomial over the field rational functions finite $\mathbb{F}_q$ characteristic $p$, then proportion $n$-tuples $(t_1,\ldots, t_n)$ monic polynomials degree $d$ which $f(t_1,\ldots, t_n,X)$ reducible out all $O(dq^{-d/2})$.