## Publicacions Matemàtiques

- Publ. Mat.
- Volume 56, Number 1 (2012), 65-89.

### Isolated singularities of binary differential equations of degree $n$

T. Fukui and J. J. Nuño-Ballesteros

#### 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 $D_3$.

#### Article information

**Source**

Publ. Mat., Volume 56, Number 1 (2012), 65-89.

**Dates**

First available in Project Euclid: 15 December 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.pm/1323972967

**Mathematical Reviews number (MathSciNet)**

MR2918184

**Zentralblatt MATH identifier**

1282.37017

**Subjects**

Primary: 37C15: Topological and differentiable equivalence, conjugacy, invariants, moduli, classification

Secondary: 34C20: Transformation and reduction of equations and systems, normal forms 34A34: Nonlinear equations and systems, general 53A07: Higher-dimensional and -codimensional surfaces in Euclidean n-space 53A60: Geometry of webs [See also 14C21, 20N05]

**Keywords**

Totally real differential form principal lines Darbouxian umbilics index

#### Citation

Fukui, T.; Nuño-Ballesteros, J. J. Isolated singularities of binary differential equations of degree $n$. Publ. Mat. 56 (2012), no. 1, 65--89. https://projecteuclid.org/euclid.pm/1323972967