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We give a general upper bound for the irrationality exponent of algebraic Laurent series with coefficients in a finite field. Our proof is based on a method introduced in a different framework by Adamczewski and Cassaigne. It makes use of automata theory and, in our context, of a classical theorem due to Christol. We then introduce a new approach which allows us to strongly improve this general bound in many cases. As an illustration, we give a few examples of algebraic Laurent series for which...

Decimal expansions of classical constants such as $\sqrt{2}$, $\pi$ and $\zeta \left(3\right)$ have long been a source of difficult questions. In the case of finite characteristic numbers (Laurent series with coefficients in a finite field), where no carry-over difficulties appear, the situation seems to be simplified and drastically different. On the other hand, the theory of Drinfeld modules provides analogs of real numbers such as $\pi$, $e$ or $\zeta$ values. Hence, it became reasonable to enquire how “complex” the Laurent representation...

En 1976, Baum et Sweet ont donné le premier exemple d’une série formelle algébrique de degré $3$ sur ${𝔽}_2\left(T\right)$ ayant un développement en fraction continue dont les quotients partiels sont tous des polynômes en $T$ de degré $1$ ou $2$. Cette série formelle est l’unique solution dans le corps ${𝔽}_2\left(\left(T^{-1}\right)\right)$ de l’équation $T{X}^3+X-T=0$. En 1986, Mills et Robbins ont décrit un algorithme permettant de calculer le développement en fraction continue de la série de Baum et Sweet. Dans cet article, nous considérons les équations plus...