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Cylinders Through Five Points: Complex and Real Enumerative Geometry
Author

Daniel Lichtblau
Organization: Wolfram Research, Inc.
Conference

ADG 2006
Conference location

Pontevedra, Spain
Description

This is from work presented at ADG 2006, Pontevedra, Spain, August 30, 2006.


Abstract
It is known that five points in R^3 generically determine a finite number of cylinders containing those points. We discuss ways in which it can be shown that the generic (complex) number of solutions, with multiplicity, is six, of which an even number will be real valued and hence correspond to actual cylinders in R^3. We partially classify the case of no real solutions in terms of the geometry of the five given points. We also investigate the special case where the five given points are coplanar, as it differs from the generic case for both complex and real valued solution cardinalities.
Subjects

*Mathematica Technology > Programming > Symbolic Computation
*Mathematics > Algebra > Polynomials
*Mathematics > Geometry > Computational Geometry
*Mathematics > Geometry > Surfaces
Keywords

Enumerative geometry, Gröbner bases, nonlinear systems, constraint geometry
URL

http://webs.uvigo.es/adg2006/
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ADG2006_talk.pdf (195 KB) - PDF Document
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cylinder_5_points_enumeration.pdf (558.9 KB) - PDF Document
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ADG2006_talk.nb (883.8 KB) - Mathematica Notebook
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cylinder_5_points_enumeration.nb (2.1 MB) - Mathematica Notebook