Explicit Hilbert's Irreducibility Theorem in Function Fields

TL;DR

This paper provides a quantitative version of Hilbert's irreducibility theorem for function fields over finite fields, showing how often specialized polynomials remain irreducible as degrees grow.

Contribution

It establishes an explicit bound on the proportion of reducible specializations of multivariate polynomials over finite fields, extending Hilbert's irreducibility theorem to a quantitative setting.

Findings

Proves a bound of O(dq^{-d/2}) on reducible specializations

Quantifies irreducibility behavior over finite fields

Extends classical Hilbert's theorem to a quantitative form

Abstract

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 of rational functions over a finite field $\mathbb{F}_q$ of characteristic $p$, then the proportion of $n$-tuples $(t_1,\ldots, t_n)$ of monic polynomials of degree $d$ for which $f(t_1,\ldots, t_n,X)$ is reducible out of all $n$-tuples of degree $d$ monic polynomials is $O(dq^{-d/2})$.