Let be a finite extension of a global field. Such an extension can be generated over by a single element. The aim of this article is to prove the existence of a ”small” generator in the function field case. This answers the function field version of a question of Ruppert on small generators of number fields.
Let be an algebraic subvariety of a torus and denote by the complement in of the Zariski closure of the set of torsion points of . By a theorem of Zhang, is discrete for the metric induced by the normalized height . We describe some quantitative versions of this result, close to the conjectural bounds, and we discuss some applications to study of the class group of some number fields.
Nous étudions les propriétés arithmétiques des itérés de certains automorphismes polynomiaux affines. Nous traitons des questions concernant les points périodiques et non-périodiques, en particulier nous comptons les points rationnels dans les orbites des points non-périodiques. Nous traitons le cas des automorphismes réguliers et triangulaires. Nous achevons de répondre aux questions en dimension 2 et montrons que la situation est nettement plus compliquée en dimension supérieure.