It is well known that an irreducible algebraic curve is rational (i.e. parametric) if and only if its genus is zero. In this paper, given a tolerance ǫ > 0 and an ǫ-irreducible algebraic affine plane curve C of proper degree d, we introduce the notion of ǫ-rationality, and we provide an algorithm to parametrize approximately affine ǫ-rational plane curves, without exact singularities at infinity, by means of linear systems of (d − 2)-degree curves. The algorithm outputs a rational parametrization of a rational curve C of degree at most d which has the same points at infinity as C. Moreover, although we do not provide a theoretical analysis, our empirical analysis shows that C and C are close in practice.

The linear complete differential resultant of a finite set of linear ordinary differential polynomials is defined. We study the computation by linear complete differential resultants of the implicit equation of a system of n linear differential polynomial parametric equations in n − 1 differential parameters. We give necessary conditions to ensure properness of the system of differential polynomial parametric equations.

Let K be an ordinary differential field with derivation ∂. Let P be a system of n linear differential polynomial parametric equations in n − 1 differential parameters with implicit ideal ID. Given a nonzero linear differential polynomial A in ID we give necessary and sufficient conditions on A for P to be n − 1 dimensional. We prove the existence of a linear perturbation P φ of P so that the linear complete differential resultant ∂CRes φ associated to P φ is nonzero. A nonzero linear differential polynomial in ID is obtained from the lowest degree term of ∂CRes φ and used to provide an implicitization algorithm for P.

This is the author’s\ud version of a work that was accepted for publication in\ud Journal of Symbolic Computation. Changes resulting from the publishing\ud process, such as peer review, editing, corrections,\ud structural formatting, and other quality control mechanisms may not be\ud reflected in this document.\ud Changes may have been made to this work since it was submitted for\ud publication.\ud A definitive version was subsequently published in Journal of Symbolic\ud Computation vol. 56 pp. 80-106 (2013).\ud DOI: 10.1016/j.jsc.2013.04.002We present an algorithm that, given a non-rational irreducible\ud real space curve, satisfying certain conditions, computes a rational\ud parametrization of a space curve near the input one. For a given\ud tolerance \epsilon > 0, the algorithm checks whether a planar projection\ud of the given space curve is \epsilon -rational and, in the affirmative\ud case, generates a planar parametrization that is lifted to a space\ud parametrization. This output rational space curve is of the same\ud degree as the input curve, both have the same structure at infinity,\ud and the Hausdorff distance between their real parts is finite.\ud Moreover, in the examples we check that the distance is small.This work has been developed, and partially supported, by the Spanish “Ministerio de Ciencia e\ud Innovación” under the Project MTM2008-04699-C03-01, and by the “Ministerio de Economía y Competitividad”\ud under the project MTM2011-25816-C02-01. All authors belong to the Research Group\ud ASYNACS (Ref. CCEE2011/R34)

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