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Cylinders Through Five Points: Complex and Real Enumerative Geometry

  • Daniel Lichtblau
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4869)

Abstract

It is known that five points in ℝ3 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 ℝ3. 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.

Keywords

Enumerative geometry Gröbner bases nonlinear systems 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Daniel Lichtblau
    • 1
  1. 1.Wolfram Research, Inc., 100 Trade Center Dr., Champaign, IL 61820USA

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