First Steps Towards Radical Parametrization of Algebraic Surfaces

TL;DR

This paper introduces radical parametrizations for algebraic surfaces, providing algorithms for specific families and proving their stability under certain geometric transformations.

Contribution

It defines radical parametrization for surfaces, develops algorithms for various surface families, and shows these parametrizations are preserved under specific geometric operations.

Findings

Algorithms for radical parametrization of certain surface families.

Radical parametrizations are preserved under offset and conchoid constructions.

Applicable to surfaces with specific singularities and degrees.

Abstract

We introduce the notion of radical parametrization of a surface, and we provide algorithms to compute such type of parametrizations for families of surfaces, like: Fermat surfaces, surfaces with a high multiplicity (at least the degree minus 4) singularity, all irreducible surfaces of degree at most 5, all irreducible singular surfaces of degree 6, and surfaces containing a pencil of low-genus curves. In addition, we prove that radical parametrizations are preserved under certain type of geometric constructions that include offset and conchoids.