Smooth connectivity in real algebraic varieties
arxiv(2024)
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.
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