Computing the Topological Degree of Maps Between 2-Spheres

Abstract

We describe an effective method for computing the topological degree of continuous functions $R:S^2 \to S^2$, where $S^2$ is the Riemann sphere. Our approach generalizes the degree formula for rational functions of complex polynomials, $\frac{f}{g}$, without common zeros. To apply our method, it is necessary to represent the function $R$ as the ratio of two continuous complex-valued functions $f$ and $g$ without common zeros. By using the Hopf fibration, this method reduces the problem to computing the winding number of a loop. This enables us to compute the degree of $\frac{f}{g}$ even when $f$ and $g$ are arbitrary continuous complex functions without common zeros, and the fraction has a limit at infinity (which can be finite or infinite). Specifically, if $f$ and $g$ are complex polynomials in $z$ and $\bar{z}$, and the highest-degree homogeneous component of the polynomial with the greater algebraic degree has a finite or infinite limit as $|z|\to\infty$, then the problem reduces to counting the roots of a complex polynomial inside the unit circle, obtained from this component.