Computing branches and asymptotes of meromorphic functions

TL;DR

This paper introduces a new method for computing branches and asymptotes of complex curves, extending previous algorithms to handle non-rational parametrizations in higher-dimensional spaces.

Contribution

It develops a novel approach for analyzing special curves in n-dimensional space, utilizing the concept of perfect curves to generalize asymptote computation.

Findings

New algorithm for non-rational parametrized curves

Extension of asymptote computation to higher dimensions

Introduction of the concept of perfect curves

Abstract

In this paper, we first summarize the existing algorithms for computing all the generalized asymptotes of a plane algebraic curve implicitly or parametrically defined. From these previous results, we derive a method that allows to easily compute the whole branch and all the generalized asymptotes of a special curve defined in n-dimensional space by a parametrization that is not necessarily rational. So, some new concepts and methods are established for this type of curves. The approach is based on the notion of perfect curves introduced from the concepts and results presented in previous papers.