Smooth connectivity in real algebraic varieties

TL;DR

This paper introduces algorithms for analyzing the connectivity and topological features of smooth real algebraic varieties, using gradient paths to determine connected components and smooth connectivity.

Contribution

It adapts existing methods to compute connectivity and Euler characteristic of smooth parts of real algebraic varieties, including singularity analysis.

Findings

Algorithms successfully compute connected components and Euler characteristic.

Gradient ascent/descent paths effectively determine smooth connectivity.

Examples demonstrate practical applicability of the approach.

Abstract

A standard question in real algebraic geometry is to compute the number of connected components of a real algebraic variety in affine space. By adapting an approach for determining connectivity in complements of real hypersurfaces by Hong, Rohal, Safey El Din, and Schost, algorithms are presented for computing the number of connected components, the Euler characteristic, and deciding the connectivity between two points for a smooth manifold arising as the complement of a real hypersurface of a real algebraic variety. When taking such real hypersurface to be the set of singular points, this yields an approach for determining smooth connectivity in a real algebraic variety. The method is based upon gradient ascent/descent paths on the real algebraic variety and several examples are included to demonstrate the approach.