5
$\begingroup$

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?

$\endgroup$

1 Answer 1

7
$\begingroup$

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.)

$\endgroup$
2
  • $\begingroup$ Where could I find a proof of the existence of such curves $C_i$ and maps $\phi_i$? $\endgroup$
    – 352506
    Dec 17, 2015 at 20:41
  • $\begingroup$ @352506 I think that one could take the proof described in Serre's Lectures on the Mordell-Weil Theorem and make it explicit, but offhand I don't know a reference where it is done. Hmmm... And that might just be for the $t$ such that $f(x,t)$ becomes reducible. Did you try searching on Google for "Effective Hilbert Irreducibility Theorem"? Among the top entries was the following, which likely has what you want: Effective Hilbert irreducibility, Erich Kaltofen, Information and Control, Volume 66, Issue 3, September 1985, Pages 123-137. $\endgroup$ Dec 17, 2015 at 21:01

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.