Local Parametrization of Space Curves at Singular Points

García, María Emilia Alonso; Mora, Teo; Niesi, Gianfranco; Raimondo, Mario Di

doi:10.1007/978-3-642-77586-4_5

Cited by 14 publications

(20 citation statements)

References 7 publications

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“…In the first part of the paper (Sections 2, 3 and 4), we consider C an irreducible real algebraic space curve over the field of complex numbers C implicitly defined by two irreducible polynomials f 1…”

Section: Introductionmentioning

confidence: 99%

Asymptotes of space curves

Blasco

Pérez–Díaz

2015

Journal of Computational and Applied Mathematics

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In this paper, we generalize the results presented in [4] for the case of real algebraic space curves. More precisely, given an algebraic space curve C implicitly defined, we show how to compute the generalized asymptotes. In addition, we show how to deal with this problem for the case of a given curve C parametrically defined. The approaches are based on the notion of approaching curves introduced in [5].

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Section: Introductionmentioning

confidence: 99%

Asymptotes of space curves

Blasco

Pérez–Díaz

2015

Journal of Computational and Applied Mathematics

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“…. , σ ϵ (ϕ n )(t)) is another solution of the system, where [27]). We refer to these solutions as the conjugates of ϕ.…”

Section: Previous Results For Implicit Space Curvesmentioning

confidence: 97%

Characterizing the finiteness of the Hausdorff distance between two algebraic curves

Blasco

Pérez–Díaz

2015

Journal of Computational and Applied Mathematics

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a b s t r a c tIn this paper, we present a characterization for the Hausdorff distance between two given algebraic curves in the n-dimensional space (parametrically or implicitly defined) to be finite. The characterization is related with the asymptotic behavior of the two curves and it can be easily checked. More precisely, the Hausdorff distance between two curves C and C is finite if and only if for each infinity branch of C there exists an infinity branch of C such that the terms with positive exponent in the corresponding series are the same, and reciprocally.

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“…This algebraic curve is introduced as follows. Let F be as in (1). We consider y and y as independent variables, let us say y and z.…”

Section: Preliminariesmentioning

confidence: 99%

Solving First Order Autonomous Algebraic Ordinary Differential Equations by Places

Falkensteiner

Sendra

2019

Math.Comput.Sci.

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Given a first-order autonomous algebraic ordinary differential equation, we present a method for computing formal power series solutions by means of places. We provide an algorithm for computing a full characterization of possible initial values, classified in terms of the number of distinct formal power series solutions extending them. In addition, if a particular initial value is given, we present a second algorithm that computes all the formal power series solutions, up to a suitable degree, corresponding to it. Furthermore, when the ground field is the field of the complex numbers, we prove that the computed formal power series solutions are all convergent in suitable neighborhoods. keyword Algebraic autonomous differential equation, algebraic curve, local parametrization, place, formal power series solution, analytic solution.

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Local Parametrization of Space Curves at Singular Points

Cited by 14 publications

References 7 publications

Asymptotes of space curves

Asymptotes of space curves

Characterizing the finiteness of the Hausdorff distance between two algebraic curves

Solving First Order Autonomous Algebraic Ordinary Differential Equations by Places

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