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Rational algebraic curves have been intensively studied in the last decades, both from the theoretical and applied point of view. In applications (e.g. level curves, linear homotopy deformation, geometric constructions in computer aided design, image detection, algebraic differential equations, etc.), there often appear unknown parameters. It is possible to adjoin these parameters to the coefficient field as transcendental elements. In some particular cases, however, the curve has a different behavior than in the generic situation treated in this way. In this paper, we show when the singularities and thus the (geometric) genus of the curves might change. More precisely, we give a partition of the affine space, where the parameters take values, so that in each subset of the partition the specialized curve is either reducible or its genus is invariant. In particular, we give a Zariski-closed set in the space of parameter values where the genus of the curve under specialization might decrease or the specialized curve gets reducible. For the genus zero case, and for a given rational parametrization, a finer partition is possible such that the specialization of the parametrization parametrizes the specialized curve. Moreover, in this case, the set of parameters where Hilbert's irreducibility theorem does not hold can be identified. We conclude the paper by illustrating these results by some concrete applications. [ABSTRACT FROM AUTHOR]

In this paper we discuss applications of our earlier work in studying certain Galois groups and splitting fields of rational functions in Q (X 0 (N)) using Hilbert's irreducibility theorem and modular forms. We also consider computational aspect of the problem using MAGMA and SAGE. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider some versions of a game when two players Nora and Wanda in some order are choosing the coefficients of a degree d polynomial. The aim of Nora is to get a polynomial which has no roots in some field or, more generally, is irreducible over that field or, even more generally, has the largest possible Galois group S d , while the aim Wanda is the opposite. We show that in order to obtain an irreducible polynomial for Nora it suffices to have the last move. However, to ensure that the splitting field of the resulting polynomial with integer coefficients has Galois group S d Nora needs to have at least three moves for each even d ≥ 4 . For d = 4 we show that Nora can always get the Galois group S 4 if Nora starts and they play alternately. [ABSTRACT FROM AUTHOR]

We prove a result about reducibility behavior of Thue polynomials over the rationals that was conjectured in [7] . More precisely, we show that, apart from few explicitly given exceptions, these polynomials have only finitely many reducible integer specializations. Special cases have been proved e.g. by Müller in [7] , Theorem 4.9, and Langmann [6, Satz 3.5] . The proof uses ramification theory to reduce the assertion to a statement about permutation groups containing an n -cycle. This statement is finally proven with the help of the classification of primitive permutation groups containing an n -cycle (a result which rests on the classification of finite simple groups). [ABSTRACT FROM AUTHOR]

The central theme of the paper is the specialization of algebraic function field extensions. Our main results are Tchebotarev type theorems for Galois function field extensions, finite or infinite, over various base fields: under some conditions, we extend the classical finite field case to number fields, p -adic fields, PAC fields, function fields κ ( x ) , etc. We also compare the Tchebotarev conclusion – existence of unramified local specializations with Galois group any cyclic subgroup of the generic Galois group (up to conjugation) – to the Hilbert specialization property. For a function field extension with the Tchebotarev property, the exponent of the Galois group is bounded by the l.c.m. of the local specialization degrees. Local–global questions arise for which we provide answers, examples and counter-examples. [ABSTRACT FROM AUTHOR]

The central topic of this paper is this question: is a given k- étale algebra l El/k the specialization of a given k-cover f : X →B of the same degree at some unramified point t0 ∊B(k)? We reduce it to finding krational points on a certain k-variety, which we then study over various fields k of diophantine interest: finite fields, local fields, number fields, etc. We have three main applications. The first one is the following Hilbert-Grunwald statement. If f : X → P1 is a degree n Q-cover with monodromy group Sn over Q, and finitely many suitably large primes p are given with partitions {dp,1, . . ., dp,sp } of n, there exist infinitely many specializations of f at points t0∊Q that are degree n field extensions with residue degrees dp,1, . . ., dp,sp at each prescribed prime p. The second one provides a description of the separable closure of a PAC field k of characteristic p= 2: it is generated by all elements y such that ym - y ∊k for some m = 2. The third one involves Hurwitz moduli spaces and concerns fields of definition of covers. [ABSTRACT FROM AUTHOR]

We prove diophantine inequalities involving various distances between two distinct algebraic points of an algebraic curve. These estimates may be viewed as extensions of classical Liouville''s inequality. Our approach is based on a transcendental construction using algebraic functions. Next we apply our results to Hilbert''s irreducibility Theorem and to some classes of diophantine equations in the circle of Runge''s method. [Copyright &y& Elsevier]