## $$p$$-adic and real subanalytic sets.(English)Zbl 0693.14012

Denote by $${\mathbb{Z}}_ p\{X\}$$ the ring of restricted power series over $${\mathbb{Z}}_ p$$ in the set of variables $$X_ 1,\dots,X_ m$$. Let $$P_ n$$ be the subset of $${\mathbb{Q}}_ p^*$$ consisting of $$n$$-th powers. A basic subset of $${\mathbb{Z}}_ p^ m$$ is a subset of the type $$B=\{x\in {\mathbb{Z}}_ p^ m;\quad f(x)=0,\quad g_ 1(x)\in P_{n(1)},\dots,g_ r(x)\in P_{n(r)}\}$$ where $$f,g_ 1,\dots,g_ r\in {\mathbb{Z}}_ p\{X\}$$ and n(1),…,n(r) are positive integers. A D-function $$f:\quad {\mathbb{Z}}_ p^ m\to {\mathbb{Z}}_ p$$ is a function obtained by composing maps induced by restricted power series and the map $$D:\quad {\mathbb{Z}}_ p^ 2\to {\mathbb{Z}}_ p$$ which is defined by $$D(x,y)=x/y$$ if $$| x| \leq | y|$$ and $$y\neq 0$$ and $$D(x,y)=0$$ otherwise. Similarly as basic subsets one defines D-basic subsets. The central result of the paper is the
Elimination theorem: Let $$\pi:\quad {\mathbb{Z}}_ p^ m\to {\mathbb{Z}}_ p^ n$$ for $$m\geq n$$ be a projection. Then the image of a basic subset B in $${\mathbb{Z}}_ p^ m$$ is D-basic in $${\mathbb{Z}}_ p^ n.$$
A subset S of a $$p$$-adic manifold M is called semianalytic if each point x of M admits an open neighborhood U of x such that $$U\cap S$$ is a finite union of sets of the type $$\{y\in U;\quad f(y)=0,\quad g_ 1(y)\in P_{n(1)},\dots,g_ r(y)\in P_{n(r)}\}$$ where $$f,g_ 1,\dots,g_ r$$ are analytic functions on U and n(1),…,n(r) are positive integers. A subset S of M is called subanalytic if for each point x of M there is an open neighborhood U of x such that $$U\cap S$$ is an image of a semianalytic subset $$S'$$ of $$U\times {\mathbb{Z}}_ p^ m$$ under the projection to U.
Uniformization theorem: Let $$S\subset {\mathbb{Z}}_ p^ m$$ be subanalytic. Then there exists a compact $$p$$-adic manifold M of dimension m and an analytic map h: $$M\to {\mathbb{Z}}_ p^ m$$ such that $$h^{-1}(S)$$ is semianalytic and such that h is a composition of finitely many blowing- ups of closed submanifolds.
As applications of the uniformization theorem the authors obtain the following results:
Rationality of Poincaré series: Let $$S\subset {\mathbb{Z}}_ p^ m$$ be subanalytic and denote by N(n,S) the cardinality of the residue classes mod $$p^ n$$ which admit representatives in S. Then $$P_ S(T)=\sum_{n\in {\mathbb{N}}}N(n,S)T^ n$$ is rational.
Rationality of Łojasiewicz exponents: Let $$S\subset {\mathbb{Z}}_ p^ m$$ be closed and subanalytic and let f,g: $$S\to {\mathbb{Z}}_ p$$ be functions whose graphs are subanalytic such that $$| f|,| g|: S\to {\mathbb{R}}$$ are continuous. Suppose that $$g^{-1}(0)\subset f^{-1}(0)$$. For $$\alpha >0$$ consider the property $P(\alpha):\quad | f(x)|^{\alpha}\leq c| g(x)| \quad for\quad some\quad c\in {\mathbb{R}}\quad and\quad all\quad x\in S.$ Then there exists some $$\alpha >0$$ such that P($$\alpha)$$ is satisfied. Let $$\alpha_ 0$$ be the infimum over all $$\alpha >0$$ such that P($$\alpha)$$ is satisfied. Then $$\alpha_ 0$$ is a rational number and if $$\alpha_ 0>0$$ then $$P(\alpha_ 0)$$ is satisfied.
Reviewer: W.Lütkebohmert

### MSC:

 14Pxx Real algebraic and real-analytic geometry 14G20 Local ground fields in algebraic geometry
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