Approximate Parametrization of Plane Algebraic Curves by Linear Systems of Curves

TL;DR

This paper introduces a method to approximately parametrize affine plane algebraic curves that are nearly rational, using linear systems of curves, and demonstrates its practical effectiveness through empirical analysis.

Contribution

It defines the concept of epsilon-rationality and provides an algorithm for approximate parametrization of epsilon-irreducible curves using linear systems.

Findings

Algorithm successfully parametrizes approximate rational curves.

Empirical results show the approximations are close in practice.

Method works without exact singularities at infinity.

Abstract

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 $\epsilon>0$ and an $\epsilon$-irreducible algebraic affine plane curve $\mathcal C$ of proper degree $d$, we introduce the notion of $\epsilon$-rationality, and we provide an algorithm to parametrize approximately affine $\epsilon$-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 $\bar{\mathcal C}$ of degree at most $d$ which has the same points at infinity as $\mathcal C$. Moreover, although we do not provide a theoretical analysis, our empirical analysis shows that $\bar{\mathcal C}$ and $\mathcal C$ are close in practice.