Theorem on six vertices of a plane curve via the Sturm theory

1995/10/26 by Laurent Guieu, L. Guieu, Guieu, L. +5
Arts and Humanities · Mathematics · #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Historical Studies and Socio-cultural Analysis #Point processes and geometric inequalities #dg-ga #math.DG

paper · pdf · doi:10.48550/arxiv.dg-ga/9510008

10 pages, 4 Postscript figures

arxiv created 1995/10/26 · openalex publication_date 1995/10/26 · arxiv updated 2016/08/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We discuss the theorem on the existence of six points on a convex closed plane curve in which the curve has a contact of order six with the osculating conic. (This is the ``projective version'' of the well known four vertices theorem for a curve in the Euclidean plane.) We obtain this classical fact as a corollary of some general Sturm-type theorems.

Related

• The Four Vertex Theorem and its Converse
• A discrete analog of Segre's theorem on spherical curves
• Projective geometry of polygons and discrete 4-vertex and 6-vertex theorems
• Sextactic points on the Fermat cubic curve and arrangements of conics
• The Hesse pencil of plane curves and osculating conics