Analyzing the Topology Types arising in a Family of Algebraic Curves Depending On Two Parameters

TL;DR

This paper introduces an algorithm to analyze how the topology of algebraic plane curves varies with two parameters, extending previous methods from one-parameter families to more complex two-parameter cases.

Contribution

It generalizes existing algorithms from one-parameter to two-parameter algebraic curve families, enabling detailed topology analysis across parameter space.

Findings

Algorithm computes a finite partition of parameter space

Topology remains invariant within each partition element

Extends previous one-parameter methods to two-parameter families

Abstract

Given the implicit equation $F(x,y,t,s)$ of a family of algebraic plane curves depending on the parameters $t,s$, we provide an algorithm for studying the topology types arising in the family. For this purpose, the algorithm computes a finite partition of the parameter space so that the topology type of the family stays invariant over each element of the partition. The ideas contained in the paper can be seen as a generalization of the ideas in \cite{JGRS}, where the problem is solved for families of algebraic curves depending on one parameter, to the two-parameters case.