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

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.