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Zbl 0867.32011
Schoutens, Hans
Uniformization of rigid subanalytic sets. (English)
[J] Compos. Math. 94, No.3, 227-245 (1994). ISSN 0010-437X; ISSN 1570-5846/e

The main theorem of this paper is a uniformization theorem for the (rigid) strongly subanalytic sets introduced by {\it H. Schoutens} [ibid., 269-295 (1994; see the following review)]. It says that after finitely many local blow-ups, a strongly subanalytic subset of an affinoid manifold becomes semianalytic (we are restricted to fields of characteristic zero here). The proof uses Hironaka's resolution of singularities together with the explicit representation (by quantifier elimiation techniques) of a strongly subanalytic set obtained by Schoutens in the above-cited paper. A strongly subanalytic subset of a reduced affinoid variety $\text{Sp }A$ may be represented by a finite system of inequalities in absolute value among strongly $D$-functions over $A$. A strongly $D$-function over $A$ is obtained by iterating composition of overconvergent power series over $A$ with functions $D(f,g)$, where $f$ and $g$ are $D$-functions and $D(a,b) =a/b$ if $|a,b|\leq 1$, else $D(a,b)=0$. At the first stage of complexity, a $D$-function is given by $D(f,g)$, where $f$ and $g$ are elements of $A$. After enough local blow-ups, one of $f$, $g$ divides the other in a Zariski-open set, thus permitting replacement of the function $D(f,g)$ by an analytic function. To prove uniformization, one iterates this operation, successively eliminating $D$'s in favor of analytic functions, eventually obtaining a semianalytic set. This result is a generation of the uniformization theorem for subanalytic sets over $\bbfQ_p$ obtained by {\it J. Denef} and {\it L. P. D. van den Dries} [Ann. Math., II. Ser. 128, No. 1, 79-138 (1988; Zbl 0693.14012)].\par In addition to this, Schoutens shows that a strongly subanalytic set in a (reduced) affinoid variety is the image of a semianalytic set of a rigid analytic variety under a proper analytic map. Thus, Schoutens' class of rigid subanalytic sets is a close analogue of the class of real subanalytic sets. The main idea here is roughly that a semianalytic set of a unit polydisk ${\bold B}^n$ defined using overconvergent power series is even semianalytic in the proper variety ${\bold P}^n$.\par Finally, Schoutens shows that if $S$ is a strongly subanalytic subset of the rigid analytic variety $N$ and if $\varphi:N \to M$ is a proper map into the quasi-compact rigid analytic variety $M$, then there is some integer $A$ such that for all $x\in M$, $\#(S\cap \varphi^{-1}(x))\in \{1,\dots,A\}\cup \{\infty\}$.
[Zachary Robinson (MR 96b:32042)]

MSC 2000:: *32P05 Non-Archimedean complex analysis
32B20 Semi-analytic sets, etc.
14G20 p-adic ground fields
14P15 Real analytic and semianalytic sets
03C10 Quantifier elimination and related topics

Keywords: uniformization; rigid subanalytic sets

Citations: Zbl 0867.32013; Zbl 0693.14012

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