Asymptotes and perfect curves

Abstract

We develop a method for computing all the generalized asymptotes of a real plane algebraic curve C implicitly defined by an irreducible polynomial f(x, y) ∈ R[x, y]. The approach is based on the notion of perfect curve introduced from the concepts and results presented in Blasco and P´erez-D´ıaz (2013). In addition, we study some properties concerning perfect curves and in particular, we provide a necessary and sufficient condition for a plane curve to be perfect. Finally, we show that the equivalent class of generalized asymptotes for a branch of a plane curve can be described as an affine space R m for a certain m.