Cylinders Through Five Points: Complex and Real Enumerative Geometry

It is known that five points in 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 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. 2000 Mathematics Subject Classification: 14N10, 14Q10, 51N20, 52A15, 65C05, 65H10, 68U20. Key words and phrases: Enumerative geometry, Gröbner bases, nonlinear systems, constraint geometry. This is a slightly updated version of a paper that appears in: Proceedings of the Sixth International Workshop on Automated Deduction in Geometry (ADG 2006), Francisco Botana and Tomas Recio, editors. Lecture Notes in...

• Type: Conference
• Year: 2006
• Where: ADG
• Authors: Daniel Lichtblau
• Keywords: Actual Cylinders, ADG 2006, Finite Number, Mathematics, Real Valued Solution