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**Rigid subanalytic sets.**
*(English)*
Zbl 0792.14010

\(K\) denotes an algebraically closed field which is complete with respect to a non-archimedean valuation. Let \(R=\{x \in K \mid | x | \leq 1\}\) and \(p=\{x \in K \mid | x |<1\}\). The paper develops a theory of subanalytic sets in \(R^ m \times p^ n\) and an analytic language for \((R,p,| K |)\). This is an analogue of the theory of real subanalytic sets and it extends work of J. Denef and L. van den Dries for local fields.

The main result is the elimination of quantifiers in this language. The starting point is the introduction of a ring of analytic functions on \(R^ m \times p^ n\), closely related to the Tate-algebras. These rings satisfy the Weierstrass theorems and are noetherian. The analytic language is build from this ring of functions and two other functions \(D_ 0\), \(D_ 1\).

Semi-analytic subsets of \(R^ m \times p^ n\) are given by equalities, norm inequalities and strict norm inequalities involving the rings of functions on \(R^ m \times p^ n\). Subanalytic sets are projections of semi-analytic sets. It is shown that the subanalytic subsets of \(R^ m \times p^ n\) are precisely given by the (quantifier free) formulas of the analytic language. A structure theorem for subanalytic sets is proved as well as a number of Łojasiewicz-inequalities.

The main result is the elimination of quantifiers in this language. The starting point is the introduction of a ring of analytic functions on \(R^ m \times p^ n\), closely related to the Tate-algebras. These rings satisfy the Weierstrass theorems and are noetherian. The analytic language is build from this ring of functions and two other functions \(D_ 0\), \(D_ 1\).

Semi-analytic subsets of \(R^ m \times p^ n\) are given by equalities, norm inequalities and strict norm inequalities involving the rings of functions on \(R^ m \times p^ n\). Subanalytic sets are projections of semi-analytic sets. It is shown that the subanalytic subsets of \(R^ m \times p^ n\) are precisely given by the (quantifier free) formulas of the analytic language. A structure theorem for subanalytic sets is proved as well as a number of Łojasiewicz-inequalities.

Reviewer: M.van der Put (Groningen)