Algebraic and algorithmic aspects of radical parametrizations

TL;DR

This paper introduces algebraic methods and algorithms to analyze and reparametrize varieties defined by radical parametrizations, facilitating implicitization and rational reparametrization checks.

Contribution

It presents new algebraic constructions and algorithms specifically designed for studying radical parametrizations and their reparametrizations.

Findings

Algorithms for implicitization of radical parametrizations

Methods to determine if radical parametrizations can be rationally reparametrized

Theoretical framework for algebraic analysis of radical varieties

Abstract

In this article algebraic constructions are introduced in order to study the variety defined by a radical parametrization (a tuple of functions involving complex numbers, $n$ variables, the four field operations and radical extractions). We provide algorithms to implicitize radical parametrizations and to check whether a radical parametrization can be reparametrized into a rational parametrization.