\(p\)-adic and real subanalytic sets.

*(English)*Zbl 0693.14012Denote 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.

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 |