Explicit Hilbert's Irreducibility Theorem in Function Fields

Abstract

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

10 pages