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Soient $K$ un corps de nombres de degré $n$ sur le corps des nombres rationnels $\mathbf{Q}$, $v$ une place de $K$. Nous démontrons que pour presque tout couple $(\alpha ,\beta )\in K\times \mathbf{Q}$, avec $\alpha \ne \beta$, on a $|\alpha -\beta |\>H\left(\alpha {)}^{-2n\sqrt{n}}H\right(\beta {)}^{-4\sqrt{n}}$, où $H(·)$ désigne la hauteur de Weil absolue. Un résultat semblable vaut quand le corps des approximants $\mathbf{Q}$ est remplacé par un corps de nombres quelconque.

We prove a version of the Hilbert Irreducibility Theorem for linear algebraic groups. Given a connected linear algebraic group $G$, an affine variety $V$ and a finite map $\pi :V\to G$, all defined over a finitely generated field $\kappa$ of characteristic zero, Theorem 1.6 provides the natural necessary and sufficient condition under which the set $\pi \left(V\right(\kappa \left)\right)$ contains a Zariski dense sub-semigroup $\Gamma \subset G\left(\kappa \right)$; namely, there must exist an unramified covering $p:\tilde{G}\to G$ and a map $\theta :\tilde{G}\to V$ such that $\pi \circ \theta =p$. In the case $\kappa =\mathbb{Q}$, $G={𝔾}_a$ is the additive group, we reobtain the...

We provide a lower bound for the number of distinct zeros of a sum $1+u+v$ for two rational functions $u,v$, in term of the degree of $u,v$, which is sharp whenever $u,v$ have few distinct zeros and poles compared to their degree. This sharpens the “$abcd$-theorem” of Brownawell-Masser and Voloch in some cases which are sufficient to obtain new finiteness results on diophantine equations over function fields. For instance, we show that the Fermat-type surface $x^a+y^a+z^c=1$ contains only finitely many rational or elliptic curves,...

Generalizing a result of Pourchet, we show that, if $\alpha ,\beta$ are power sums over $\mathbb{Q}$ satisfying suitable necessary assumptions, the length of the continued fraction for $\alpha \left(n\right)/\beta \left(n\right)$ tends to infinity as $n\to \infty$. This will be derived from a uniform Thue-type inequality for the rational approximations to the rational numbers $\alpha \left(n\right)/\beta \left(n\right)$, $n\in \mathbb{N}$.

We prove that there are only finitely many odd perfect powers in $\mathbb{N}$ having precisely four nonzero digits in their binary expansion. The proofs in fact lead to more general results, but we have preferred to limit ourselves to the present statement for the sake of simplicity and clarity of illustration of the methods. These methods combine various ingredients: results (derived from the Subspace Theorem) on integer values of analytic series at $S$-unit points (in a suitable $\nu$-adic convergence), Roth’s...

In our previous work we proved a bound for the $gcd(u-1,v-1)$, for $S$-units $u,v$ of a function field in characteristic zero. This generalized an analogous bound holding over number fields, proved in [3]. As pointed out by Silverman, the exact analogue does not work for function fields in positive characteristic. In the present work, we investigate possible extensions in that direction; it turns out that under suitable assumptions some of the results still hold. For instance we prove Theorems 2 and 3 below, from...