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Finiteness results for Hilbert’s irreducibility theorem. (English) Zbl 1014.12002
Let $$k$$ be a finitely generated field extension of $${\mathbb Q}$$, and let $$R$$ be a finitely generated subring of $$k$$. For a polynonial $$f(t,X)\in k(t)[X]$$, denote by $$\text{Red}_f(R)$$ the set of those $$\overline t\in R$$ for which $$f(\overline t,X)$$ is defined and reducible over $$k$$. The main purpose of the paper is to give several sufficient conditions which guarantee that $$\text{Red}_f(R)$$ is a finite set. A number of criteria for finiteness are given in terms of certain conditions on the ramification of the places $$t\mapsto \infty$$ of $$k(t)$$ in $$k(t,x)$$ where $$x$$ is a root of $$f$$. For example, it is shown that, assuming $$f$$ irreducible, $$\text{Red}_f(R)$$ is finite in each of the following cases:
(i) there exists a root $$x$$ such that the place is unramified and the Galois group of $$f(t,X)$$ over $$k(t)$$ has no subgroup of index 2;
(ii) $$k={\mathbb Q}$$, $$R={\mathbb Z}$$ and the greatest common divisor of all ramification indices of the places above $$t\mapsto \infty$$ is 1;
(iii) $$k={\mathbb Q}$$, $$R={\mathbb Z}$$ and there exists a root $$x$$ such that the place is unramified and either this place or the polynomial $$f$$ has odd degree.
These results are applied to special polynomials, such as homogeneous polynomials and polynomials of the form $$P(X)-tQ(X)$$, obtaining generalizations of previous results of K. Langmann.
A criterion of different type is the following: Assume that the Galois group of $$f(t,X)$$ over $$k(t)$$ acts doubly transitively on the roots of $$f$$. Then $$\text{Red}_f(R)$$ is finite, unless $$f$$ is absolutely irreducible and the curve $$f(t,X)=0$$ has genus 0.
A similar deduction is obtained for $$\text{Red}_f(k)$$, but under the strongest assumption that the curve $$f(t,X)=0$$ has genus $$> 1$$. Under the weaker condition that the Galois group is primitive, a certain condition on the composition factors implies again the finiteness of $$\text{Red}_f(R)$$.
For $$k={\mathbb Q}$$, $$R={\mathbb Z}$$, and polynomials of odd prime degree, the polynomials of type $$f(t,X)=h(X)-t$$ have the property that $$\text{Red}_f(R)$$ is infinite, and an argument is given to show that these are essentially the only examples of this type.
The main tool for all the preceding results is Siegel’s theorem about algebraic curves having infinitely many integral points in a number field. Using the classification of simple groups, the author adds some other sufficient conditions on the Galois group of $$f(t,X)$$ over $$k(t)$$ which guarantee that $$\text{Red}_f(R)$$ is finite.

##### MSC:
 12E25 Hilbertian fields; Hilbert’s irreducibility theorem 12E30 Field arithmetic 14H25 Arithmetic ground fields for curves 20B15 Primitive groups 20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
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