Nonorientable polynomial foliations on the plane

“…Now, suppose that f ∈ R [x, y] is a polynomial of degree n and consider on the xy-plane the two fields of asymptotic directions, X 1 and X 2 , defined by equation (2). Similarly, as V. Guíñez does in [9], we shall prove that the induced quadratic differential forms s * 1 (II f ) and s * 2 (II f ) can be extended to an analytical quadratic differential form defined on the sphere.…”

Section: Projection Into the Poincaré Spherementioning

confidence: 53%

“…where a, b, c ∈ R [x, y] are polynomials of degree at most n, the function b 2 − 4ac is nonnegative at every point of the xy-plane and b 2 − 4ac vanishes at a point p if and only if a, b, c vanish simultaneously at p. He extends the foliations determined by equation (1) to the line at infinity and proves, among other things, that the topological behaviour of these foliations in a neighbourhood of a singular point at infinity, is one of the types shown in Fig. 1 (see [9], Remark 2.9). When f ∈ R[x, y] is a polynomial, the second fundamental form II f is a polynomial quadratic differential form that, in general, is not positive: there are disjoint open sets on the plane where the discriminant of this form is negative.…”

Section: Introductionmentioning

confidence: 90%

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On the geometric structure of certain real algebraic surfaces

Guadarrama-García

Ortiz-Rodríguez

2017

Geom Dedicata

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In this paper we study the affine geometric structure of the graph of a polynomial f ∈ R[x, y]. We provide certain criteria to determine when the parabolic curve is compact and when the unbounded component of its complement is hyperbolic or elliptic. We analyse the extension to the real projective plane of both fields of asymptotic lines and the Poincaré index at its singular points at infinity. We exhibit an index formula for the field of asymptotic lines involving the number of connected components of the projective Hessian curve of f and the number of godrons. As an application of this investigation, we obtain upper bounds, respectively, for the number of godrons having an interior tangency and when they have an exterior tangency.

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“…In a broader context, there is the investigation of singular points, their index and topological type, that arise at infinity as a result of the projective extension of a quadratic differential form on the plane [10]. The fields of principal directions of a surface, denoted by X k , k = 1, 2, are the zero set of a quadratic form called the form of principal curvatures.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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

Guilfoyle,

Ortiz-Rodríguez

2018

Preprint

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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 Poincaré-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.

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Nonorientable polynomial foliations on the plane

Cited by 10 publications

References 6 publications

Positive quadratic differential forms: linearization, finite determinacy and versal unfolding

Positive quadratic differential forms: linearization, finite determinacy and versal unfolding

On the geometric structure of certain real algebraic surfaces

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

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