Exceptional specializations of Galois groups in the Hilbert Irreducibility Theorem

Question

Suppose $f(x,t)\in\mathbb{Q}(t)[x]$ is an irreducible polynomial with Galois group G. For any rational number $a$ we may consider the polynomial $f(x,a)\in\mathbb Q[x]$ and its corresponding Galois group $G_a$, which is a subgroup of $G$. By the Hilbert Irreducibility Theorem, the groups $G$ and $G_a$ are the same outside a thin set of values of $a$. (Here I'm using "thin" in the sense of Serre.) If I have a specific polynomial $f$, how explicitly can the set of exceptional values of $a$ (those for which $G_a$ is a proper subgroup of $G$) be described? Are there any references that discuss this question?

Answer 1

In some sense it can be described quite explicitly. There is a finite set of curves $C_i$ and maps $\phi_i:C_i\to\mathbb P^1$ of degree at least $2$ defined over $\mathbb Q$ such that the desired thin set is contained in the union $$ \bigcup_{i=1}^n \phi_i\bigl(C_i(\mathbb Q)\bigr).$$ And I'm pretty sure that one can effectively determine $C_1,\ldots,C_n$ and $\phi_1,\ldots,\phi_n$ from the polynomial $f(x,t)$, although in practice finding specific equations might be a rather hard commutative algebra calculation with Grobner bases, etc.

Having said that, there is still a major problem, namely we need to determine $C_i(\mathbb Q)$ for $1\le i\le n$. For those $C_i$ of genus $0$, this can be done using (1) $C_i(\mathbb Q)\ne\emptyset$ if and only if $C_i(\mathbb Q_p)\ne\emptyset$ for all completions, including $p=\infty$; (2) Hensel's lemma; (3) if $C_i(\mathbb Q)\ne\emptyset$, then one can use the known point to find a parametrization $C_i(\mathbb Q)\cong\mathbb P^1(\mathbb Q)$. But for those $C_i$ of genus $g\ge1$, we do not have effective algorithms for determining the rational points. So for a particular $f(x,t)$, you might be lucky and be able to completely describe the thin set, but in general I do not believe that there is an effective algorithm. On the other hand, there are effective upper bounds for $\#C(\mathbb Q)$ when $g\ge2$, so one might be able to write the thin set as an explicitly given infinite set plus an unknown finite set whose size is explicitly bounded. (I'd have to think a bit more about the genus 1 curves.)