Parametric equations of plane sextic curves with a maximal set of double points

TL;DR

This paper provides explicit parametric equations for all irreducible plane sextic curves with at most double points and maximal total Milnor number, including minimal field and coefficient size considerations.

Contribution

It explicitly constructs parametric equations for all such sextic curves with maximal Milnor number, optimizing for minimal field degree and coefficient size.

Findings

Explicit parametrizations for all relevant sextic curves

Minimal field degree for parametrizations identified

Coefficients optimized for simplicity

Abstract

We give explicit parametric equations for all irreducible plane projective sextic curves which have at most double points and whose total Milnor number is maximal (is equal to 19). In each case we find a parametrization over a number field of the minimal possible degree and try to choose coordinates so that the coefficients are as small as we can do.