Galois groups over rational function fields and explicit Hilbert irreducibility

TL;DR

This paper develops methods to compute Galois groups of bivariate polynomials over rational function fields and explicitly identify exceptional specializations where the Galois group or factorization differs from the generic case.

Contribution

It introduces new techniques for determining Galois groups over rational function fields and explicitly describing exceptional specializations, with applications to arithmetic dynamics.

Findings

Methods for computing Galois groups over $\, ext{Q}(t)$

Explicit descriptions of exceptional specializations

Application to a new result in arithmetic dynamics

Abstract

Let $P\in\mathbb Q[t,x]$ be a polynomial in two variables with rational coefficients, and let $G$ be the Galois group of $P$ over the field $\mathbb Q(t)$. It follows from Hilbert's Irreducibility Theorem that for most rational numbers $c$ the specialized polynomial $P(c,x)$ has Galois group isomorphic to $G$ and factors in the same way as $P$. In this paper we discuss methods for computing the group $G$ and obtaining an explicit description of the exceptional numbers $c$, i.e., those for which $P(c,x)$ has Galois group different from $G$ or factors differently from $P$. To illustrate the methods we determine the exceptional specializations of three sample polynomials. In addition, we apply our techniques to prove a new result in arithmetic dynamics.