Quelques aspects sur la géométrie des surfaces algébriques réelles

Ortiz-Rodríguez, Adriana

doi:10.1016/s0007-4497(03)00007-1

Cited by 10 publications

(11 citation statements)

References 13 publications

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“…This number is an upper bound for the number of Gaussian cusps on the generic algebraic surfaces of degree n in RP 3 . For the graph of a real polynomial of degree n in the plane, the second author has obtained an upper bound for the number of Gaussian cusps: 2(n − 2)(5n − 12), [9]. In Theorem 5 we obtain a refinement to this bound, namely, (n − 2)(5n − 12).…”

Section: Introductionmentioning

confidence: 91%

On the affine geometry of the graph of a real polynomial

2012

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Given a surface defined as the graph of a real polynomial in two variables, we analyze some basic subsets characterized by its tangential singularities. If the parabolic curve is compact we provide certain criteria to determine when the unbounded component of its complement is hyperbolic. Moreover, we obtain an upper bound of the number of Gaussian cusps that holds even if the parabolic curve is non compact.

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Section: Introductionmentioning

confidence: 91%

On the affine geometry of the graph of a real polynomial

2012

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“…In [19], A. Ortiz-Rodríguez proved, among other things, that for any real factorisable polynomial of degree n, f = ℓ i , the following holds: (i) The lines ℓ i are the only components of the flecnodal curve of the graph of f , (ii) This graph has exactly n(n − 2) godrons, and Proof. Theorem 6 provides an alternative and very simple proof of Proposition 3: Since the flecnodal curve consists of straight lines, at each godron g the asymptotic tangent line and the asymptotic curve coincide with one of such straight lines.…”

Section: Theoremmentioning

confidence: 99%

“…(Theorem 1 and Lemma 13 of[19]) All godrons of the graph of a real factorisable polynomial are negative.…”

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confidence: 99%

A Projective Invariant for Swallowtails and Godrons, and Global Theorems on the Flecnodal Curve

Uribe-Vargas¹

2006

MMJ

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We show some generic (robust) properties of smooth surfaces immersed in the real 3-space (Euclidean, affine or projective), in the neighbourhood of a godron (called also cusp of Gauss): an isolated parabolic point at which the (unique) asymptotic direction is tangent to the parabolic curve. With the help of these properties and a projective invariant that we associate to each godron we present all possible local configurations of the flecnodal curve at a generic swallowtail in R 3 . We present some global results, for instance: In a hyperbolic disc of a generic smooth surface, the flecnodal curve has an odd number of transverse self-intersections (hence at least one self-intersection).

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“…Arnold's problem [Arn04, 2001-2] on the topology of the parabolic curve asks in particular for the maximal number of components of P(P ) or at least for it asymptotic (See also problem 2000-2). Ortiz-Rodriguez constructed in [OR03] smooth real algebraic surfaces of any degree d ≥ 3 whose parabolic curve is smooth and has d(d−1)(d−2) 2 connected components.…”

Section: Introductionmentioning

confidence: 99%

“…The maximal number of compact connected components of such a curve in this special case is the Date: November 19, 2018. subject of problem [Arn04, 2001-1] (See also problem 2000-1). In [OR03], Ortiz-Rodriguez constructed real polynomials Q(X, Y ) of degree d ≥ 3 whose Hessian define smooth real curves with (d−1)(d−2) 2 compact connected components in R 2 . In small degrees, this construction has been slightly improved in [ORS07].…”

Section: Introductionmentioning

confidence: 99%