$g_R'$ that the sum of the roots of $g_R'(z)$ is $$\frac{n}{n+1} (R - a_{n-1}),$$ where $a_{n-1}$ is the coefficient of $z^{n-1}$ in $f(z)$. Since the sum $\beta_1+\cdots+\beta_{n-1}$ is bounded, so is $$\beta_n(R) - \frac{n}{n+1} R = -\left(\frac{n}{n+1}a_{n-1}+\beta_1+\cdots+\beta_{n-1}\right).$$ That is, $\beta_n(R)$ is within a bounded distance of $\frac{n}{n+1}R$, and so we have $$\arg(\beta_n) \rightarrow 0, \qquad \arg(\beta_n-R) \rightarrow \pi, \qquad \text{and} \qquad
   |\beta_n|\rightarrow \infty,$$ whence $$\arg(g_R(\beta_n)) = \arg((\beta_n-R)f(\beta_n)) \rightarrow \pi \,.$$

For $R$ sufficiently large, $g_R(z)$ has no repeated roots. By Lemma \[lemma:singularities\], we see that $\arg(g_R(\beta_i(R)))$ becomes arbitrarily close to $\arg(f(\alpha_i))+ \pi$ for $i<n$, and to $\pi$ for $i=n$. Using the hypotheses that ${C_{\theta}}(f)$ is nonsingular and $\theta \neq 0$, it follows that $\arg(g_R(\beta_i(R)))$ is not congruent to $\theta \pmod{\pi}$ for any $i$, and so $C_{\theta}(g_R)$ is nonsingular for $R$ sufficiently large.

The next step in proving Theorem \[thm:matchings\] is to show that the topological behavior of the curve ${C_{\theta}}(g_R)$ inside the disk $D_r$ is the same as that of the curve ${C_{\theta}}(f)$; that is, the passage from $f(z)$ to $g_R(z)$ preserves the combinatorial type of the matching ${M}(f,\theta)$.

\[prop:disk\] Let $f(z)$ be a monic polynomial of degree $n$, let $\theta \in
(0,\pi)$ be such that ${C_{\theta}}(f)$ is nonsingular, and 