# Connected components of real $CB_{n}$ algebraic varieties

## TL;DR

This paper studies the connected components of real algebraic varieties invariant under the $CB_{n}$-Coxeter group, providing bounds, geometric characterizations, and a decomposition of the space of such varieties.

## Contribution

It introduces a new method to analyze the geometry of $CB_{n}$-algebraic varieties using Cerf-Douady theory and constructs $CB_{n}$-polynomials via Young-posets, establishing bounds on connected components.

## Key findings

- Maximum number of connected components can reach $2^{n}+1$ for specific coefficients.
- Provides a decomposition of the space of $CB_{n}$-algebraic varieties.
- Establishes bounds and geometric characterizations using advanced topological theories.

## Abstract

Connected components of real algebraic varieties invariant under the $CB_{n}$-Coxeter group are investigated. In particular, we consider their maximal number and their geometric and topological properties. This provides a decomposition for the space of $CB_{n}$-algebraic varieties. We construct $CB_{n}$-polynomials using Young-posets and partitions of integers. Our results establish bounds on the number of connected components for a given set of coefficients. It turns out that this number can achieve an upper bound of $2^{n}+1$ for specific coefficients. We introduce a new method to characterize the geometry of these real algebraic varieties, using J. Cerf and A. Douady theory for varieties with angular boundary and the theory of chambers and galleries. We provide several examples that bring out the essence of these results.