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

TL;DR

This paper investigates the global behavior of principal direction fields on polynomial surfaces, deriving a formula relating umbilic point indices to polynomial factors and analyzing umbilic points at infinity.

Contribution

It introduces a Poincaré-Hopf type formula linking umbilic point indices to polynomial factors and characterizes the topology of umbilic points at infinity.

Findings

The sum of indices depends on the number of real linear factors of the highest degree part.

Umbilic points at infinity are isolated with index 1/2.

Topological type of infinite umbilic points is a Lemon.

Abstract

The global qualitative behaviour of fields of principal directions for the graph of a real valued polynomial function $f$ on the plane are studied. We provide a Poincar\'e-Hopf type formula where the sum over all indices of the principal directions at its umbilic points only depends upon the number of real linear factors of the homogeneous part of highest degree of $f$. Moreover, we study the projective extension of these fields and prove, under generic conditions, that every umbilic point at infinity of these extensions is isolated, has index equal to 1/2 and its topological type is a Lemon.