there is exactly one point of $I \cap S_r$ lying between each point of $R \cap S_r$, each arc $R_i$ must have an *odd* number of points of $I \cap S_r$ on either side of it. It follows that some arc $I_j$ joins a point on one side of $R_i$ with a point on the other, and therefore must intersect $R_i$. These $n$ intersection points, one for each $R_i$, are the roots of our polynomial $f$. (This step requires only the Intermediate Value Theorem, rather than the Jordan Curve Theorem, because ${\mathop{\rm Re}\nolimits}f(z)$ changes sign along the arc $I_j$.) This establishes:

\[prop:nonsing\] Let $f(z)$ be a monic polynomial such that the curves $R$ and $I$ are nonsingular. Then $f(z)$ has $n$ roots.

Gersten and Stallings [@Gersten] complete the proof of the Fundamental Theorem of Algebra by proving that if $f(z)$ has no roots, then $f(z)$ can be perturbed to obtain a new polynomial that still has no roots but for which $R$ and $I$ are nonsingular. We give a different approach.

Observe that the curve $I$ has $2n$ asymptotes, with slopes at angles $k\pi/n$ for $k=0,\ldots,2n-1$. Similarly, $R$ has asymptotes with slopes at angles $(k + \frac{1}{2})\pi/n$. More generally, consider 