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Isolated singularities of binary differential equations of degree $n$

T. Fukui and J. J. Nuño-Ballesteros

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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

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

First available in Project Euclid: 15 December 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

Totally real differential form principal lines Darbouxian umbilics index


Fukui, T.; Nuño-Ballesteros, J. J. Isolated singularities of binary differential equations of degree $n$. Publ. Mat. 56 (2012), no. 1, 65--89.

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