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Many algorithms for determining properties of semi-algebraic sets rely upon the ability to compute smooth points [1]. We present a simple procedure based on computing the critical points of some well-chosen function that guarantees the computation of smooth points in each connected bounded component of a real atomic semi-algebraic set. Our technique is intuitive in principal, performs well on previously difficult examples, and is straightforward to implement using existing numerical algebraic geometry software. The practical efficiency of our approach is demonstrated by solving a conjecture on the number of equilibria of the Kuramoto model for then= 4 case. We also apply our method to design an efficient algorithm to compute the real dimension of algebraic sets, the original motivation for this research.
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Smooth connectivity in real algebraic varieties
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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- PAR ID:
- 10547139
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Numerical Algorithms
- Volume:
- 100
- Issue:
- 1
- ISSN:
- 1017-1398
- Format(s):
- Medium: X Size: p. 63-84
- Size(s):
- p. 63-84
- Sponsoring Org:
- National Science Foundation
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