## Differential equations on an algebraic function field. (Équations différentielles sur un corps de fonctions algébriques.)(French)Zbl 0954.12004

Let $$\mathbb E$$ denote the completion of a rational function field $$\mathbb C_p[t]$$ in one variable over the completion $$\mathbb C_p$$ of the algebraic closure of the $$p$$-adic field. Let $$\mathbb K$$ be a finite algebraic extension of $$\mathbb E$$. Both $$\mathbb E$$ and $$\mathbb K$$ are valued fields; let $$E$$ and $$K$$ denote the corresponding residual fields.
The derivation $$\frac{d}{dt}$$ of $$\mathbb C_p[t]$$ extends to a unique continuous derivation of $$\mathbb E$$ and this latter extends uniquely to a derivation of $$\mathbb K$$; we denote this derivation of $$\mathbb K$$ by $$\frac{d}{dt}$$ as well. A Frobenius endomorphism of a valued field like $$\mathbb E$$ and $$\mathbb K$$ is a field endomorphism $$\phi$$ satisfying $$|u|\leq 1 \Rightarrow |u^p-\phi(u)|< 1$$ for all $$u$$ in the field.
The author is interested in two questions: first, whether $$|\frac{1}{n!}\frac{d^nu}{dt^n}|\leq |u|$$ for all $$u \in \mathbb K$$ and all $$n$$; and second, whether every Frobenius endomorphism of $$\mathbb E$$ can be extended to a Frobenius endomorphism of $$\mathbb K$$.
The author proves that both questions have a positive answer when the residual extension $$K \supset E$$ is separable.

### MSC:

 12H25 $$p$$-adic differential equations 12J25 Non-Archimedean valued fields

### Keywords:

Frobenius endomorphism; change of variable
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### References:

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