# A proof of Kurdyka's conjecture on arc-analytic functions

## TL;DR

This paper proves Kurdyka's conjecture that arc-symmetric semialgebraic sets are exactly the zero loci of arc-analytic semialgebraic functions, establishing a fundamental correspondence in real algebraic geometry.

## Contribution

It confirms that arc-symmetric semialgebraic sets correspond precisely to radical ideals of arc-analytic semialgebraic functions, resolving a longstanding conjecture.

## Key findings

- Arc-symmetric semialgebraic sets are zero loci of arc-analytic semialgebraic functions
- Establishes a one-to-one correspondence with radical ideals
- Proves Kurdyka's conjecture in real algebraic geometry

## Abstract

We prove a conjecture of Kurdyka stating that every arc-symmetric semialgebraic set is precisely the zero locus of an arc-analytic semialgebraic function. This implies, in particular, that arc-symmetric semialgebraic sets are in one-to-one correspondence with radical ideals of the ring of arc-analytic semialgebraic functions.