Asymptote angles of polynomic root loci

In recent publications the plotting rules of polynomially parameterized root loci have been developed for characteristic polynomials in the gain as unknown with degrees 2 and 3. Unfortunately the proof of the asymptote rule is very long and intricate. In this note a generalization of the rule of asymptote angles is addressed for any polynomial with arbitrary degree. Additionally, a simple and shorter proof of the asymptote angles is obtained through the continuous dependence of the roots of the complex characteristic polynomial with respect to a parameter. The geometry of the roots of this polynomial is investigated via a variant of Rouché’s theorem.