Resumen de Sur les équations fonctionelles p-adiques aux q-différences
Jean Paul Bézivin, Abdelbaki Boutabaa
In this paper, we study the convergence of formal power series $\Phi$ solutions of functional equations of the form $\sum^t_0 P_i(x)\Phi(\varphi^{[i]}(x))=\tau(x)$ where the base field is the field $\mathbb{C}_p$ of $p$-adic numbers for a prime number $p$, and $\varphi{[k]}$ denotes the $k$-th iterate of the function $\varphi$. The case when the base field is $\mathbb{C}$ has been studien in a previous paper.\newline As an application, we prove that a formal power series solution of a system of a $q_1$-difference equation and of a $q_2$-difference equation is the Taylor series at the origin of a rational function, when the base field is the field of algebraic numbers and $q_1,q_2$ multiplicatively independents, or for a base field $K$, commutative with zero characteristic and $q_1,q_2$ algebraically independents.
