Distance Properties of ∈-Points on Algebraic Curves

This paper deals with some mathematical objects that the authors have named ∈-points (see [8]), and that appear in the problem of parametrizing approximately algebraic curves. This type of points are used as based points of the linear systems of curves that appear in the parametrization algorithms, and they play an important role in the error analysis. In this paper, we focus on the general study of distance properties of ∈-points on algebraic plane curves, and we show that if P⋆ is an ∈-point on a plane curve C of proper degree d, then there exists an exact point P on C such that its distance to P⋆ is at most \(\sqrt \varepsilon\) if P⋆ is simple, and O(\(\sqrt \varepsilon\)1/2d) if P⋆ is of multiplicity r > 1. Furthermore, we see how these results particularize to the univariate case giving bounds that fit properly with the classical results in numerical analysis.