---
abstract: 'In this paper, we introduce the notion of infinity branches as well as approaching curves. We present some properties which allow us to obtain an algorithm that compares the behavior of two implicitly defined algebraic plane curves at the infinity. As an important result, we prove that if two plane algebraic curves have the same asymptotic behavior, the Hausdorff distance between them is finite.'
author:
- |
    Angel Blasco and Sonia Pérez-Díaz\
    Departamento de Física y Matemáticas\
    Universidad de Alcalá\
    E-28871 Madrid, Spain\
    angel.blasco@uah.es, sonia.perez@uah.es
title: Asymptotic Behavior of an Implicit Algebraic Plane Curve
---

[**Keywords:**]{} Implicit Algebraic Plane Curve; Infinity Branches; Convergent Branches; Asymptotic Behavior; Approaching Curves

Introduction
============

Unirational algebraic varieties, play an important role in the frame of practical applications (see [@HSW] and [@HL97]). In particular, many authors have studied different problems related to plane algebraic curves that are defined implicitly (see e.g. [@SWP] and [@Walker]). In this paper, we deal with the notion of infinity branches which is a very important tool to analyze the behavior of an implicitly defined algebraic plane curve at the infinity. For instance, determining the infinity branches of an implicit real algebraic plane curve is an important step in 