On the irreducible components of globally defined semianalytic sets

TL;DR

This paper introduces amenable $C$-semianalytic sets in real analytic manifolds, establishing their properties and developing a theory of irreducible components that generalizes several classical concepts in algebraic and analytic geometry.

Contribution

It defines amenable $C$-semianalytic sets and develops a theory of irreducibility and components, extending classical notions to this broader setting.

Findings

Amenable $C$-semianalytic sets have well-behaved Zariski closures.

A natural definition of irreducibility for these sets is established.

The theory generalizes concepts from algebraic, analytic, Nash, and semialgebraic 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.