Strictly analytic functions on p-adic analytic open sets

Abstract

Let $K$ be an algebraically closed complete ultrametric field. M. Krasner and P. Robba defined theories of analytic functions in $K$, but when $K$ is not spherically complete both theories have the disadvantage of containing functions that may not be expanded in Taylor series in some disks. On other hand, affinoid theories are only defined in a small class of sets (union of affinoid sets) \cite{2}, \cite{13} and \cite{17}. Here, we suppose the field $K$ topologically separable (example $\Bbb C_p$). Then, we give a new definition of strictly analytic functions over a large class of domains called analoid sets. Our theory uses the notion of $T$-sequence which caracterizes analytic sets in the sense of Robba. Thereby we obtain analytic functions satisfying the property of analytic continuation and which, however, will admit expansion in power series (resp. Laurent series) in any disk (resp. in any annulus). Moreover, the algebra of analytic functions will be stable by derivation. The process consists of defining a large class of analytic sets $D$, and a class of admissible sets making a covering of such a $D$, so that we obtain a sheaf on $D$. We finally give an example of differential equation whose solutions are strictly analytic functions in an analoid set. Such an example might not be involved in theories based on affinoid sets.