of higher degree. Indeed, regarding a monic polynomial of degree $n$ as a vector in ${{\mathbb R}}^{2n}$ by taking the real and imaginary parts of its coefficients, the subset of ${{\mathbb R}}^{2n}$ for which at least one of the curves $R$ and $I$ is singular is an algebraic set. Its complement, the set of polynomials for which ${M}(f,0)$ and ${M}(f,\pi/2)$ are both defined, is therefore a real semialgebraic set. However, every connected component of a real semialgebraic set is again semialgebraic [@Bochnak Thm. 2.4.5]; since the set of polynomials which yield a particular pair of matchings is a union of such connected components, that set is itself semialgebraic.

Combinatorics of basketballs {#section:basketball}
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Throughout, if $a\leq b$ are integers, we write $[a,b]$ for the set $\{a,a+1,\dots,b-1,b\}$. (It should be clear from context whether this notation refers to an interval in ${{\mathbb Z}}$ or in ${{\mathbb R}}$.)

Let $a\leq b$ be integers. A [*partition*]{} of $[a,b]$ is a collection of pairwise disjoint sets (called [*blocks*]{}) whose union is $[a,b]$. The number of blocks is the [*order*]{} of the partition. Two blocks are said to [*cross*]{} if there are integers $i<j<k<\ell$ such that $i,k$ belong to one block and $j,\ell$ belong to 