Resumen de Applications de la théorie de Nevanlinna p-adique

Abdelbaki Boutabaa

• Let $p$ be a prime number. We note $\mathbb{C}_p$ the completion of $\mathbb{Q}_p$ and $M(\mathbb{C}_p)$ the space of meromorphic functions in all $\mathbb{C}_p$. Let $P(x,y,y',\cdots,y^{(n)})$ be a differential polynomial with coefficients in $\mathbb{C}_p(x)$ and let be $R(x,y)\in\mathbb{C}_p(x,y)$. \newline \begin{center} We prove the following: If the differential equation: \end{center} $$P(x,y,y',\cdots,y^{(n)})=R(x,y)$$ has a meromorphic solution $y=f(x)\notin\mathbb{C}_p(x)$, then $R(x,y)$ is a polynomial in $y$ with coefficients in $\mathbb{C}_p(x)$.