Radical Parametrization of Algebraic Curves by Adjoint Curves

TL;DR

This paper introduces algorithms for parametrizing algebraic curves by radicals, focusing on curves with genus up to 4 and certain degree and singularity conditions, expanding the class of curves that can be explicitly parametrized.

Contribution

It provides new algorithms for radical parametrization of algebraic curves based on genus and singularity criteria, including non-plane and plane curves of specific degrees.

Findings

Algorithms for genus ≤ 4 curves over algebraically closed fields.

Parametrization of plane curves with degree ≤ 5 and certain singularities.

Extension to non-plane algebraic curves with radical parametrizations.

Abstract

We present algorithms for parametrizing by radicals an irreducible curve, not necessarily plane, when the genus is less o equal to 4 and they are defined over an algebraically closed field of characteristic zero. In addition, we also present an algorithm for parametrizing by radicals any irreducible plane curve of degree $d$ having at least a point of multiplicity $d-r$, with $1\leq r \leq 4$ and, as a consequence, every irreducible plane curve of degree $d \leq 5$ and every irreducible singular plane curve of degree 6.