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Smooth connectivity in real algebraic varieties

https://doi.org/10.1007/s11075-024-01952-3

Cummings, Joseph; Hauenstein, Jonathan_D; Hong, Hoon; Smyth, Clifford_D (October 2024, Numerical Algorithms)

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.
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Improving angular speed uniformity of rational parameterization using piecewise radical reparameterization

https://doi.org/10.1007/s10472-025-09976-8

Hong, Hoon; Wang, Dongming; Yang, Jing (March 2025, Annals of Mathematics and Artificial Intelligence)

Free, publicly-accessible full text available March 27, 2026
Parametric “non-nested” discriminants for multiplicities of univariate polynomials

https://doi.org/10.1007/s11425-023-2143-3

Hong, Hoon; Yang, Jing (March 2024, Science China Mathematics)

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