Abstract
A quadratic point on a surface in ℝP3 is a point at which the surface can be approximated by a quadric abnormally well (up to order 3). We conjecture that the least number of quadratic points on a generic compact nondegenerate hyperbolic surface is 8; the relation between this and the classic Carathéodory conjecture is similar to the relation between the six-vertex and the four-vertex theorems on plane curves. Examples of quartic perturbations of the standard hyperboloid confirm our conjecture. Our main result is a linearization and reformulation of the problem in the framework of the 2-dimensional Sturm theory; we also define a signature of a quadratic point and calculate local normal forms recovering and generalizing the Tresse-Wilczynski theorem.
Original language | English (US) |
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Pages (from-to) | 178-193 |
Number of pages | 16 |
Journal | Proceedings of the Steklov Institute of Mathematics |
Volume | 258 |
Issue number | 1 |
DOIs | |
State | Published - Sep 2007 |
All Science Journal Classification (ASJC) codes
- Mathematics (miscellaneous)