Topological Behavior of Families of Algebraic Curves Continuously Depending on a Parameter Under Certain Conditions.

Abstract

En trabajos previos del autor, se considera el problema de determinar los tipos topológicos en una familia de curvas algebraicas planas, algebraicamente dependientes de un parámetro. En este trabajo, estos resultados se generalizan, bajo ciertas condiciones, al caso de una familia de curvas algebraicas planas dependientes de forma continua de un parámetro t, que toma valores en un subconjunto de la recta real que es unión de una cantidad finita de intervalos abiertos. Los resultados conducen al cálculo de un polinomio R(t) con la propiedad de que para todo intervalo contenido en U, que no contenga ninguna raíz de R(t), el tipo topológico de la familia no varía. Un caso importante en el que los resultados son aplicables, es el caso en que los coeficientes de las curvas son algebraicamente independientes. Si el número de raíces de R(t) es finito, los tipos topológicos presentes en la familia pueden calcularse mediante métodos bien conocidos.

In previous works of the author, the question of computing the different shapes arising in a family of algebraic curves algebraically depending on a real parameter was addressed. In this work we show how the ideas in these papers can be used to extend the results to a more general class of families of algebraic curves, namely families not algebraically but just continuously depending on a parameter. These families correspond to polynomials in the variables x,y whose coefficients are continuous functions of a parameter t taking values in U, where U is in general the union of finitely many open intervals. Under certain conditions, here we provide an algorithm for computing a univariate real function R(t), with the property that the topology of the family stays invariant along every real interval I contained in U, and not containing any real root of R(t). In that situation, a partition of the real line where each element gives rise to a same shape arising in the family, can be computed. Then, these shapes can be described by using well-known methods. An important situation when the method is applicable is the case when the coefficients are algebraically independent, or can be expressed in terms of algebraically independent functions.