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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.

A rational a ne parametrization of an algebraic surface establishes a rational correspondence of the a ne plane with the surface. We consider the problem of computing the degree of such a rational map. In general, determining the degree of a rational map can be achieved by means of elimination theoretic methods. For curves, it is shown that the degree can be computed by gcd computations. In this paper, we show that the degree of a rational map induced by a surface parametrization can be computed by means of gcd and univariate resultant computations. The basic idea is to express the elements of a generic ÿbre as the ÿnitely many intersection points of certain curves directly constructed from the parametrization, and deÿned over the algebraic closure of a ÿeld of rational functions.Let P( t ) be a rational a ne parametrization of a unirational surface V over an algebraically closed ÿeld K of characteristic zero. Associated with the parametrization P( t ), we have the rational map P : K 2 → V ; t → P( t ), where P (K 2 ) ⊂ V is dense.

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