Ephraim's Pencils

TL;DR

This paper investigates the topological behavior of a family of complex analytic curves defined by a parameterized equation, establishing bounds on special parameter values and applying results to critical values at infinity of polynomials.

Contribution

It provides a bound on the number of special parameters for which the topological type changes and applies this to estimate critical values at infinity of complex polynomials.

Findings

Number of nonzero special values does not exceed the number of components of the curve

Most curves in the family share the same topological type

Application to estimate critical values at infinity of complex polynomials

Abstract

Let $f(x,y)=0$ and $l(x,y)=0$ be respectively a singular and a regular analytic curve defined in the neighborhood of the origin of the complex plane. We study the family of analytic curves $f(x,y)-tl(x,y)^M=0$, where $t$ is a complex parameter. For all but a finite number of parameters the curves of this family have the same embedded topological type. The exceptional parameters are called special values. We show that the number of nonzero special values does not exceed the number of components of the curve $f(x,y)=0$ counted without multiplicities. Then we apply this result to estimate the number of critical values at infinity of complex polynomials in two variables.