## Abstract

A quadratic point on a surface in ℝP^{3} 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)