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We prove a quantitative version of Hilbert's irreducibility theorem for function fields: If $f(T_1,\ldots, T_n,X)$ is an irreducible polynomial over the field of rational functions over a finite field $\mathbb{F}_q$ of characteristic $p$, then the proportion of $n$-tuples $(t_1,\ldots, t_n)$ of monic polynomials of degree $d$ for which $f(t_1,\ldots, t_n,X)$ is reducible out of all $n$-tuples of degree $d$ monic polynomials is $O(dq^{-d/2})$.

We prove new results on the distribution of rational points on ramified covers of abelian varieties over finitely generated fields $k$ of characteristic zero. For example, given a ramified cover $\pi : X \to A$, where $A$ is an abelian variety over $k$ with a dense set of $k$-rational points, we prove that there is a finite-index coset $C \subset A(k)$ such that $\pi(X(k))$ is disjoint from $C$. Our results do not seem to be in the range of other methods available at present; they confirm predictions coming from Lang's conjectures on rational points, and also go in the direction of an issue raised by Serre regarding possible applications to the Inverse Galois Problem. Finally, the conclusions of our work may be seen as a sharp version of Hilbert's irreducibility theorem for abelian varieties., Comment: 53 pages. Minor corrections. Added remarks 1.5, 7.4, 7.9 and 7.10. Final version

In this note, we show that, for any f ∈ ℤ x and any prime number p , there exists g ∈ ℤ x for which the polynomial f x − g x p is irreducible over ℚ . For composite p ≥ 2 , this assertion is not true in general. However, it holds for any integer p ≥ 2 if f is not of the form a h x k , where a ≠ 0 and k ≥ 2 are integers and h ∈ ℤ x .

In this paper we obtain new quantitative forms of Hilbert's Irreducibility Theorem. In particular, we show that if $f(X, T_1, \ldots, T_s)$ is an irreducible polynomial with integer coefficients, having Galois group $G$ over the function field $\mathbb{Q}(T_1, \ldots, T_s)$, and $K$ is any subgroup of $G$, then there are at most $O_{f, \varepsilon}(H^{s-1+|G/K|^{-1}+\varepsilon})$ specialisations $\mathbf{t} \in \mathbb{Z}^s$ with $|\mathbf{t}| \le H$ such that the resulting polynomial $f(X)$ has Galois group $K$ over the rationals.

The central topic is this question: is a given $k$-\'etale algebra $\prod_lE_l/k$ the specialization of a given $k$-cover $f:X\rightarrow B$ at some point $t_0\in B(k)$? Our main tool is a {\it twisting lemma} that reduces the problem to finding $k$-rational points on a certain $k$-variety. Previous forms of this twisting lemma are generalized and unified. New applications are given: a Grunwald form of Hilbert's irreducibility theorem over number fields, a non-Galois variant of the Tchebotarev theorem for function fields over finite fields, some general specialization properties of covers over PAC or ample fields.

Let X be a projective curve defined over \(\mathbb{Q}\) and \(t \in \mathbb{Q}(X)\) a non-constant rational function of degree ν ≥ 2. For every \(n \in \mathbb{Z}\) pick \(P_{n} \in X(\bar{\mathbb{Q}})\) such that t(P n ) = n. A result of Dvornicich and Zannier implies that, for large N, among the number fields \(\mathbb{Q}(P_{1}),\ldots, \mathbb{Q}(P_{N})\) there are at least cN∕ logN distinct; here, c > 0 depends only on the degree ν and the genus g = g(X). We prove that there are at least N∕(logN)1−η distinct fields, where η > 0 depends only on ν and g.

In the context of Hilbert’s irreducibility theorem, it is an open question whether there exists a bound for the least hilbertian specialization in N that is polynomial in the degree d and the logarithmic height log(H) of the polynomial P (T,Y ) in question. A positive answer would be useful, notably for algorithmic applications. We obtain a polynomial bound in log(H) and dHi(P) where Hi(P) — the Hilbert index of P — is a pure group-theoretical invariant we define and which we show to be absolutely bounded for many classes of polynomials. We also discuss further questions related to effectiveness in Hilbert’s irreducibility theorem. 2000 MSC. Primary 12E25, 14G05 ; Secondary 11C08, 12E05

The minimal ramification problem may be considered as a quantitative version of the inverse Galois problem. For a nontrivial finite group $G$, let $m(G)$ be the minimal integer $m$ for which there exists a $G$-Galois extension $N/\mathbb{Q}$ that is ramified at exactly $m$ primes (including the infinite one). So, the problem is to compute or to bound $m(G)$.In this paper, we bound the ramification of extensions $N/\mathbb{Q}$ obtained as a specialization of a branched covering $\unicode[STIX]{x1D719}:C\rightarrow \mathbb{P}_{\mathbb{Q}}^{1}$. This leads to novel upper bounds on $m(G)$, for finite groups $G$ that are realizable as the Galois group of a branched covering. Some instances of our general results are: $$\begin{eqnarray}1\leqslant m(S_{k})\leqslant 4\quad \text{and}\quad n\leqslant m(S_{k}^{n})\leqslant n+4,\end{eqnarray}$$ for all $n,k>0$. Here $S_{k}$ denotes the symmetric group on $k$ letters, and $S_{k}^{n}$ is the direct product of $n$ copies of $S_{k}$. We also get the correct asymptotic of $m(G^{n})$, as $n\rightarrow \infty$ for a certain class of groups $G$.Our methods are based on sieve theory results, in particular on the Green–Tao–Ziegler theorem on prime values of linear forms in two variables, on the theory of specialization in arithmetic geometry, and on finite group theory.

Let $X$ be a smooth and projective variety such that $X$ is fibred over the projective space. We give sufficient conditions ensuring that the fibres contain adelic points satisfying Manin-like conditions.

R´´ Nousla construction et le comptage, pour tout couple d'entiers m,n > 1, des corps de nombres de degre n dont le groupe des classes possede un "grand" m-rang. Notre technique repose essentiellement sur le theoreme d'irreductibilite de Hilbert et sur des resultats concernant les points entiers de degre borne sur des courbes. Abstract. We study the problem of constructing and enumer- ating, for any integers m,n > 1, number fields of degree n whose ideal class groups have "large" m-rank. Our technique relies fun- damentally on Hilbert's irreducibility theorem and results on in- tegral points of bounded degree on curves.

We prove that if K is a finite extension of Q , P is the set of prime numbers in Z that remain prime in the ring R of integers of K , f , g ∈ K [ X ] with deg g >deg f and f , g are relatively prime, then f + pg is reducible in K [ X ] for at most a finite number of primes p ∈ P . We then extend this property to polynomials in more than one indeterminate. These results are related to Hilbert's irreducibility theorem.

Letf (X, t)eℚ[X, t] be an irreducible polynomial. Hilbert’s irreducibility theorem asserts that there are infinitely manyt0eℤ such thatf (X, t0) is still irreducible. We say thatf (X, t) isgeneral if the Galois group off (X, t) over ℚ(t) is the symmetric group in its natural action. We show that if the degree off with respect toX is a prime ≠ 5 or iff is general of degree ≠ 5, thenf (X, t0) is irreducible for all but finitely manyt0eℤ unless the curve given byf (X, t)=0 has infinitely many points (x0,t0) withx0eℚ,t0eℤ. The proof makes use of Siegel’s theorem about integral points on algebraic curves, and classical results about finite groups, going back to Burnside, Schur, Wielandt, and others.

The simplest case of Hilbert’s Irreducibility Theorem asserts that if F (t, x) is irreducible over Q, then there exists t∗ ∈ Q such that F (t∗, x) is irreducible over Q. Many different proofs have been given for this theorem, namely Hilbert’s (1892) [H], Mertens’s (1911) [Me], Skolem’s (1921) [Sk], Dorge’s (1927) [Do], Siegel’s (1929) [Si], Eichler’s (1939) [Ei], Inaba’s (1943) [In], Fried’s (1974) [Fr], Roquette’s (1975) [Ro], Cohen’s (1981) [Co], Sprindžuk’s (1981) [Spr], Debes’s (1986) [De1], (1993) [De2]. Only the last of the quoted papers explicitly mentions the problem of estimating the size of a t∗ with the above property in terms of the degree and height of F . By the height of F , to be abbreviated H(F ), we mean the maximum absolute value of the coefficients of a constant multiple of F that has coprime integer coefficients. Debes gives actually an estimate valid for several polynomials Fi. His result reads (see Cor. 3.7 of [De2]): Let F1, . . . , Fh be irreducible polynomials in Q[t, x] such that degFi ≤ D and H(Fi) ≤ H (1). Then there exists a rational number t∗ = u/v such that each Fi(t∗, x) is irreducible over Q and

The paper has three main applications. The first one is this Hilbert-Grunwald statement. If $f:X\rightarrow \Pp^1$ is a degree $n$ $\Qq$-cover with monodromy group $S_n$ over $\bar\Qq$, and finitely many suitably big primes $p$ are given with partitions $\{d_{p,1},..., d_{p,s_p}\}$ of $n$, there exist infinitely many specializations of $f$ at points $t_0\in \Qq$ that are degree $n$ field extensions with residue degrees $d_{p,1},..., d_{p,s_p}$ at each prescribed prime $p$. The second one provides a description of the se-pa-ra-ble closure of a PAC field $k$ of characteristic $p\not=2$: it is generated by all elements $y$ such that $y^m-y\in k$ for some $m\geq 2$. The third one involves Hurwitz moduli spaces and concerns fields of definition of covers. A common tool is a criterion for an \'etale algebra $\prod_lE_l/k$ over a field $k$ to be the specialization of a $k$-cover $f:X\rightarrow B$ at some point $t_0\in B(k)$. The question is reduced to finding $k$-rational points on a certain $k$-variety, and then studied over the various fields $k$ of our applications.

Letf(X; T1, ...,Tn) be an irreducible polynomial overQ. LetB be the set ofb teZn such thatf(X;b) is of lesser degree or reducible overQ. Let ℱ={Fj}{Fj}j−1∞ be a Folner sequence inZn — that is, a sequence of finite nonempty subsetsFj ⊆Zn such that for eachvteZn, $$\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap (F_j + \upsilon )} \right|}}{{\left| {F_j } \right|}} = 1$$ Suppose ℱ satisfies the extra condition that forW a properQ-subvariety ofPn−An and ɛ>0, there is a neighborhoodU ofW(R) in the real topology such that $$\mathop {lim sup}\limits_{j \to \infty } \frac{{\left| {F_j \cap U} \right|}}{{\left| {F_j } \right|}}< \varepsilon $$ whereZn is identified withAn(Z). We prove $$\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap B} \right|}}{{\left| {F_j } \right|}} = 0$$ .

We prove Dirichlet's theorem for polynomial rings: Let F be a pseudo algebraically closed field. Then for all relatively prime polynomials a(X), b(X)\in F[X] and for every sufficiently large positive integer n there exist infinitely many polynomials c(X)\in F[X] such that a(X) + b(X)c(X) is irreducible of degree n, provided that F has a separable extension of degree n., Comment: final version

Thèse en cotutelle avec l'Italie (université de Udine). Composition du jury : Président : Pr. Michel WALDSCHMIDT, Université de Paris VI. Rapporteurs : Pr. Roger HEATH-BROWN, University of Oxford. Pr. Peter MÜLLER, Universität Würzburg. Examinateurs : Pr. Mohamed AYAD, Université du Littoral. Pr. Pietro CORVAJA, Università degli Studi di Udine.; Hilbert's irreducibility theorem gives the existence of a specialization preserving the irreducibility of a multivariate polynomial with rational coefficients. Effective versions have been given by P. Dèbes (1993) and by A. Schinzel and U. Zannier (1995). We discuss some attempts to improve these effective results : Dörge's method, congruence method inspired by an article of M. Fried and finally the use of a recent result of R. Heath-Brown about rational points on curves. This last attempt leads to a significant improvement of known results. We also give an application to the research of an algorithm for the factorization of bivariate polynomials.; Le théorème d'irréductibilité de Hilbert assure l'existence d'une spécialisation conservant l'irréductibilité d'un polynôme à plusieurs variables et à coefficients rationnels. Des versions effectives ont été données par P. Dèbes (1993) puis par U. Zannier et A. Schinzel (1995). Nous proposons ici diverses tentatives d'améliorer ces résultats effectifs : méthode de Dörge, méthode des congruences inspirée par un article de M. Fried et enfin une utilisation des résultats récents de R. Heath-Brown sur les points entiers d'une courbe algébrique. Cette dernière voie va nous permettre d'améliorer significativement les résultats connus. On finira par une application à la recherche d'un algorithme polynomial pour la factorisation d'un polynôme à deux indéterminées.

Thèse en cotutelle avec l'Italie (université de Udine). Composition du jury : Président : Pr. Michel WALDSCHMIDT, Université de Paris VI. Rapporteurs : Pr. Roger HEATH-BROWN, University of Oxford. Pr. Peter MÜLLER, Universität Würzburg. Examinateurs : Pr. Mohamed AYAD, Université du Littoral. Pr. Pietro CORVAJA, Università degli Studi di Udine.; Hilbert's irreducibility theorem gives the existence of a specialization preserving the irreducibility of a multivariate polynomial with rational coefficients. Effective versions have been given by P. Dèbes (1993) and by A. Schinzel and U. Zannier (1995). We discuss some attempts to improve these effective results : Dörge's method, congruence method inspired by an article of M. Fried and finally the use of a recent result of R. Heath-Brown about rational points on curves. This last attempt leads to a significant improvement of known results. We also give an application to the research of an algorithm for the factorization of bivariate polynomials.; Le théorème d'irréductibilité de Hilbert assure l'existence d'une spécialisation conservant l'irréductibilité d'un polynôme à plusieurs variables et à coefficients rationnels. Des versions effectives ont été données par P. Dèbes (1993) puis par U. Zannier et A. Schinzel (1995). Nous proposons ici diverses tentatives d'améliorer ces résultats effectifs : méthode de Dörge, méthode des congruences inspirée par un article de M. Fried et enfin une utilisation des résultats récents de R. Heath-Brown sur les points entiers d'une courbe algébrique. Cette dernière voie va nous permettre d'améliorer significativement les résultats connus. On finira par une application à la recherche d'un algorithme polynomial pour la factorisation d'un polynôme à deux indéterminées.

Using Moriwaki's calculation of the Q-Picard group for the moduli space of curves, I prove the strong Franchetta Conjecture in all characteristics. That is, the canonical class generates the group of rational points on the Picard scheme for the generic curve of genus g>2. Similar results hold for generic pointed curves. Moreover, I show that Hilbert's Irreducibility Theorem implies that there are many other nonclosed points in the moduli space of curves with such properties., 23 pages, major extension, to appear in Internat. J. Math

Soient k un corps de nombres, O k son anneau d'entiers et f(t, X) E k(t)[X] un polynome irreductible. Le theoreme d'irreductibilite de Hilbert fournit une infinite de specialisations entieres t → t ∈ O k telles que f(t,X) reste irreductible. Dans cet article, nous etudions l'ensemble Red f (O k ) des t ∈ O k tels que f(t,X) est reductible. Nous montrons que Red f (O k ) est un ensemble fini sous des hypotheses assez faibles. En particulier, certains de nos enonces generalisent des resultats anterieurs obtenus par des techniques d'approximations diophantiennes. Notre methode est differente. Nous utilisons de la theorie elementaire des groupes, la theorie des valuations et le theoreme de Siegel sur les points entiers des courbes algebriques. En utilisant en fait la generalisation de Siegel-Lang du theoreme de Siegel, la plupart de nos resultats sont valables sur des corps assez generaux. On peut obtenir d'autres resultats en faisant appel a la classification des groupes finis simples. Nous en donnons un apercu dans la derniere section.

Let K be an algebraic number field and IK the ring of algebraic integers in K. *K and *IK denote enlargements of K and IK respectively. Let x Є *K – K. In this paper, we are concerned with algebraic extensions of K(x) within *K. For each x Є *K – K and each natural number d, YK(x,d) is defined to be the number of algebraic extensions of K(x) of degree d within *K. x Є *K – K is called a Hilbertian element if YK(x,d) = 0 for all d Є N, d > 1; in other words, K(x) has no algebraic extension within *K. In their paper [2], P. C. Gilmore and A. Robinson proved that the existence of a Hilbertian element is equivalent to Hilbert's irreducibility theorem. In a previous paper [9], we gave many Hilbertian elements of nonstandard integers explicitly, for example, for any nonstandard natural number ω, 2ωPω and 2ω(ω3 + 1) are Hilbertian elements in *Q, where pω is the ωth prime number.

A method for obtaining very precise results along the lines of the Hilbert Irreducibility Theorem is described and then applied to a special case. In addition, the relationship of the irreducibility theorem to other tools of diophantine analysis is investigated. In particular, we give a proof of the irreducibility theorem that uses only Noether's lemma and the fact that an absolutely irreducible curve has a rational point over a finite field of large order.

In its simplest form, Hilbert’s theorem asserts : let f(t, X) be a polynomial in Q[f, X] (so in two variables), and assume that f(t, X) is irreducible. Then there exist infinitely many rational numbers t 0 such that f(t 0, X) is irreducible over Q.

We use the nonstandard methods of Chapter 13 to give a new criterion for a field K to be separably Hilbertian: there exists a nonstandard element t of an enlargement *Kof K such that t has only finitely many poles in K(t)s ⋂ *K. From this there results a second and uniform proof (Theorem 14.9) that the classical Hilbertian fields are indeed Hilbertian. In addition, a formal power series field, K 0((X 1,..., X n )) of n22652 variables over an arbitrary field K 0, is also Hilbertian (Corollary 14.18).

Hilbert’s irreducibility theorem has quite a large number of proofs, based on different principles; some give precise estimates for the number of integers that one seeks. There are several applications: the construction of elliptic curves over Q having rank ≥ 9 and the construction of extensions of Q with Galois groups S n , A n ,...