TOPOLOGY OF FAMILIES OF ALGEBRAIC CURVES CONTINUOUSLY DEPENDING ON A PARAMETER, AND APPLICATIONS

Abstract

Given a family of algebraic curves whose coefficients depend continuously on a parameter t ∈ U ⊂ ℝ (U a union of real intervals) we address the problem of computing the topology types in the family. Under certain conditions, we provide a method to compute a univariate real function R*(t), with the property that the topology of the family stays invariant along every real interval I ⊂ U not containing any real root of R*(t). So, if R*(t) has finitely many real roots the topology types in the family can be computed. We apply this result to provide bounds on the number of topology types of the family in certain cases, including the important case when the coefficients belong to a Pfaffian chain, and to analyze the existence of certain degeneracies in the real part of a surface whose family of level curves corresponds to the type studied here. The results of the paper can be seen as generalizations of previous works [Alcázar, Applications of level curves to some problems on algebraic surfaces, in Contribuciones Científicas en Honor de Mirian Andrés Gòmez, eds. Lambán, Romero and Rubio (Universidad de La Rioja, Servicio de Publicaciones, 2010), pp. 105–122, Alcázar, Schicho and Sendra, A delineability-based method for computing critical sets of algebraic surfaces, J. Symbolic Comput.42 (2007) 678–691] on the topology of families algebraically depending on a parameter t.