Smooth connectivity in real algebraic varieties

A standard question in real algebraic geometry is to compute the number of connected components of a real algebraic variety in affine space. This manuscript provides algorithms 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 considering the complement of the set of singular points of a real algebraic variety, 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 inspired by a method proposed by Hong, Rohal, Safey El Din, and Schost for complements of real hypersurfaces. Several examples are included to demonstrate the approach.