Under the assumptions of Theorem \[thm:matchings\], the curves ${C_{\theta}}(g_R)$ are nonsingular for $R$ sufficiently large.

To establish this fact, we first need the following technical result. The idea is that as $R\to\infty$, one root of $g_R'(z)$ increases without bound, while the other $n-1$ roots approach the roots of $f'(z)$.

\[lemma:singularities\] Let $\alpha_1,\ldots,\alpha_{n-1}$ denote the roots of $f'(z)$. For $R\in{{\mathbb R}}$ sufficiently large, there is an ordering $\beta_1(R),\ldots,\beta_n(R)$ of the roots of $g_R'(z)$ with the following properties:[^2]

1.  if $i \le n-1$, then $\beta_i \rightarrow \alpha_i$ as $R \rightarrow \infty$;

2.  $|\beta_n| \rightarrow \infty$ and $\arg(\beta_n)\to 0$ as $R \rightarrow \infty$;

3.  if $i \le n-1$, then $\arg(g_R(\beta_i)) \rightarrow \arg(f(\alpha_i)) + \pi$ as $R \rightarrow \infty$;

4.  $\arg(g_R(\beta_n)) \rightarrow \pi$ as $R \rightarrow \infty$.

Note that $g_R'(z) = f(z) + (z-R)f'(z)$. Put $t = 1/R$, so that $$g_R'(z)/R = (tz-1)f'(z) + tf(z).$$ Note that $\lim_{t\to 0} g'_R(z)/R=-f'(z)$. The existence of an ordering of the roots with property (1) now follows from the argument principle [@Ahlfors Theorem 18, p. 152]. Assertion (3) is evident: we have seen that $\beta_i(R)$ is bounded, and therefore that $\arg(\beta_i(R)-R) \rightarrow \pi$ as $R\to\infty$.

For (2) and (4), observe from the coefficients of $z^n$ and $z^{n-1}$ in 