in his doctoral thesis [@Gauss]. Gauss’s argument, while characteristically elegant, was itself not entirely satisfactory. Both d’Alembert’s and Gauss’s proofs were subsequently made completely rigorous (for the whole story, see, e.g., [@Stillwell pp. 195–200]).

Gauss approached the problem by examining the real algebraic curves $$\begin{aligned}
    R &= \{z {\;:~}{\mathop{\rm Re}\nolimits}f(z)=0\},\\
    I &= \{z {\;:~}{\mathop{\rm Im}\nolimits}f(z)=0\},
  \end{aligned}$$ in ${{\mathbb R}}^2$, where $f(z)$ is a complex polynomial of degree $n$. (See Figure \[basketball-example-figure\] for an example.) Note that the set of roots of $f$ is precisely $R\cap I$. For a suitably large disk $D$, each of the curves $R$ and $I$ meets the boundary circle $S=\partial
D$ in exactly $2n$ points, and it can easily be shown that the points of $R\cap S$ alternate with those of $I\cap S$ around $S$. Gauss argued that $R\cap D$ (resp. $I\cap D$) must consist of $n$ components and that each component of $R$ must cross some component of $I$ inside $D$; therefore, $\#(R \cap I)\geq n$.

We are interested in the topology of the curves $R$ and $I$, both singly and together. Much of the research on the topology of real plane algebraic curves (the topic of Hilbert’s sixteenth problem) has focused on classifying the 