Finiteness results for Hilbert's irreducibility theorem

Soient $k$ un corps de nombres, ${𝒪}_k$ son anneau d’entiers et $f(t,X)\in k\left(t\right)\left[X\right]$ un polynôme irréductible. Le théorème d’irréductibilité de Hilbert fournit une infinité de spécialisations entières $t\mapsto \overline{t}\in {𝒪}_k$ telles que $f(\overline{t},X)$ reste irréductible. Dans cet article, nous étudions l’ensemble ${\mathrm{Red}}_f\left({𝒪}_k\right)$ des $\overline{t}\in {𝒪}_k$ tels que $f(\overline{t},X)$ est réductible. Nous montrons que ${\mathrm{Red}}_f\left({𝒪}_k\right)$ est un ensemble fini sous des hypothèses assez faibles. En particulier, certains de nos énoncés généralisent des résultats antérieurs obtenus par des techniques d’approximations diophantiennes. Notre méthode est différente. Nous utilisons de la théorie élémentaire des groupes, la théorie des valuations et le théorème de Siegel sur les points entiers des courbes algébriques. En utilisant en fait la généralisation de Siegel-Lang du théorème de Siegel, la plupart de nos résultats sont valables sur des corps assez généraux. On peut obtenir d’autres résultats en faisant appel à la classification des groupes finis simples. Nous en donnons un aperçu dans la dernière section.

Let $k$ be a number field, ${𝒪}_k$ its ring of integers, and $f(t,X)\in k\left(t\right)\left[X\right]$ be an irreducible polynomial. Hilbert’s irreducibility theorem gives infinitely many integral specializations $t\mapsto \overline{t}\in {𝒪}_k$ such that $f(\overline{t},X)$ is still irreducible. In this paper we study the set ${\mathrm{Red}}_f\left({𝒪}_k\right)$ of those $\overline{t}\in {𝒪}_k$ with $f(\overline{t},X)$ reducible. We show that ${\mathrm{Red}}_f\left({𝒪}_k\right)$ is a finite set under rather weak assumptions. In particular, previous results obtained by diophantine approximation techniques, appear as special cases of some of our results. Our method is different. We use elementary group theory, valuation theory, and Siegel’s theorem about integral points on algebraic curves. Indeed, using the Siegel-Lang extension of Siegel’s theorem, most of our results hold over more general fields. Using the classification of the finite simple groups, further results can be obtained. The last section contains a short survey.

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