Isolated singularities of binary differential equations of degree $n$

Abstract

We study isolated singularities of binary differential equations of degree $n$ which are totally real. This means that at any regular point, the associated algebraic equation of degree $n$ has exactly $n$ different real roots (this generalizes the so called positive quadratic differential forms when $n=2$). We introduce the concept of index for isolated singularities and generalize Poincaré-Hopf theorem and Bendixson formula. Moreover, we give a classification of phase portraits of the $n$-web around a generic singular point. We show that there are only three types, which generalize the Darbouxian umbilics $D_1$, $D_2$ and $D_3$.