On the irreducible components of globally defined semianalytic sets

Abstract

In this work we present the concept of amenable $C$-semianalytic subset of a real analytic manifold $M$ and study the main properties of this type of sets. Amenable $C$-semianalytic sets can be understood as globally defined semianalytic sets with a neat behavior with respect to Zariski closure. This fact allows us to develop a natural definition of irreducibility and the corresponding theory of irreducible components for amenable $C$-semianalytic sets. These concepts generalize the parallel ones for: complex algebraic and analytic sets, $C$-analytic sets, Nash sets and semialgebraic sets.