Decomposition of sets in real algebraic geometry

TL;DR

This paper introduces a novel decomposition method for semialgebraic sets using arc-analytic functions, refining existing decompositions and establishing a new algebraic geometry framework with classical tools.

Contribution

It proposes a new notion of irreducibility based on arc-analytic functions, enhancing the decomposition of semialgebraic sets and developing a related algebraic geometry theory.

Findings

Refined decomposition of semialgebraic sets

Established a theory with Identity Principle and Nullstellensatz

Unified approach with Nash and rational function decompositions

Abstract

We present a new notion of decomposition of semialgebraic sets by introducing a mode of irreducibility based on arc-analytic functions. The result is a refinement of the decomposition of such sets with respect to the Zariski topology as well as a refinement of the decomposition in each of the recent approaches based on Nash and continuous rational functions. In addition, by pairing the ring of arc-analytic functions with semialgebraic sets, we obtain a theory of algebraic geometry equipped with strong tools such as the Identity Principle and the Nullstellensatz.