Localization of the complex zeros of parametrized families of polynomials

Let Pn(x)=xm+pm−1(n)xm−1+⋯+p1(n)x+pm(n) be a parametrized family of polynomials of a given degree with complex coefficients pk(n) depending on a parameter . We use Rouché’s theorem to obtain approximations to the complex roots of Pn(x). As an example, we obtain approximations to the complex roots of the quintic polynomials Pn(x)=x5+nx4−(2n+1)x3+(n+2)x2−2x+1 studied by A. M. Schöpp.