Work

Smooth connectivity in real algebraic varieties

Year: 2024

Type: article

Abstract: 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.

Source: Numerical Algorithms

DOI: 10.1007/s11075-024-01952-3

Authors: Joseph Cummings, Jonathan D. Hauenstein, Hoon Hong, Clifford D. Smyth

Institutions: University of Notre Dame, North Carolina State University, University of North Carolina at Greensboro

Language: English

Topic: Polynomial and algebraic computation

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,090

Funders: National Science Foundation, Simons Foundation, Division of Computing and Communication Foundations