Umbilic Points on the Finite and Infinite Parts of Certain Algebraic Surfaces

Authors

Brendan Guilfoyle, School of STEM, Munster Technological University, Kerry, Tralee, Co. Kerry, IrelandFollow
Adriana Ortiz-Rodríguez, Instituto de Matemáticas, Universidad Nacional Autónoma de México. Area de la Inv. Cient., Circuito exterior C.U., Mexico City 04510, México

ORCID

https://orcid.org/0000-0002-7437-5046

Document Type

Preprint

Disciplines

Algebra | Mathematics | Physical Sciences and Mathematics

Publication Details

Mathematical Proceedings of the Royal Irish Academy, vol. 123A, no. 2. © Royal Irish Academy

Abstract

The global qualitative behaviour of fields of principal directions for the graph of a real valued polynomial function f on the plane is studied. We determine and analyze the projective extension of these fields and show that they are defined by an analytic quadratic form on the whole unit 2-sphere. We prove that every umbilic point at infinity of this extension has a Poincaré-Hopf index equal to 1/2, and the topological type of a Lemon when the degree of f is 2 and the topological type of a Monstar for higher degrees. As a consequence we prove a Poincaré-Hopf type formula for the graph of f such that, if all umbilics are isolated, the sum of all indices of the principal directions at umbilic points depends only upon the number of real linear factors of the homogeneous part of highest degree of f. A similar analysis is carried out in the case of f being a homogeneous polynomial.

Recommended Citation

Guilfoyle, B., & Ortiz-Rodríguez, A. (2023). Umbilic Points on the Finite and Infinite Parts of Certain Algebraic Surfaces. Mathematical Proceedings of the Royal Irish Academy 123(2), 63-94. https://dx.doi.org/10.1353/mpr.2023.a908326.