the family of real plane curves $$\label{define-Ctheta}
  {C_{\theta}}(f) = \{ z {\;:~}{\mathop{\rm Im}\nolimits}(e^{-i\theta} f(z)) = 0\}\,,$$ parameterized by the circle ${{\mathbb R}}/\pi{{\mathbb Z}}$. We will often abuse notation by identifying $\theta\in{{\mathbb R}}/\pi{{\mathbb Z}}$ with its coset representative in the interval $[0,\pi)$. Note that ${C_{\theta}}(f)$ has $2n$ asymptotes, with slopes at angles $\frac{k\pi + \theta}{n}$ for $k=0,\ldots,2n-1$. We regard $\arg(z)$ as an element in ${{\mathbb R}}/2\pi{{\mathbb Z}}$.

The curve ${C_{\theta}}(f)$ is singular if and only if there exists $z \in {C_{\theta}}(f)$ such that $f'(z)=0$.

If $f(z) = u(x,y) + i v(x,y)$ with $u,v$ real, then ${C_{\theta}}(f)$ is the curve defined by $v(x,y) \cos\theta - u(x,y) \sin\theta = 0$. It follows that $(x_0,y_0) \in
{C_{\theta}}(f)$ is a singular point if and only if $\frac{\partial v}{\partial x} \cos\theta -
\frac{\partial u}{\partial x} \sin\theta$ and $\frac{\partial v}{\partial y} \cos\theta -
\frac{\partial u}{\partial y} \sin\theta$ vanish at $(x_0,y_0)$. By the Cauchy-Riemann equations, it follows that the partial derivatives of $u$ and $v$ with respect to $x$ and $y$ all vanish at $(x_0,y_0)$, and therefore $z_0 = x_0 + i y_0$ is a zero of $f'(z)$.

\[cor:repeated\] If $f(z)$ has no repeated roots, then the curves ${C_{\theta}}(f)$ are singular for at most $n-1$ values of $\theta$. 