  Black hole spacetimes, like the Kerr spacetime, admit both stable and
plunging orbits, separated in parameter space by the separatrix. Determining
the location of the separatrix is of fundamental interest in understanding
black holes, and is of crucial importance for modeling extreme mass-ratio
inspirals. Previous numerical approaches to locating the Kerr separatrix were
not always efficient or stable across all of parameter space. In this paper we
show that the Kerr separatrix is the zero set of a single polynomial in
parameter space. This gives two main results. First, we thoroughly analyze
special cases (extreme Kerr, polar orbits, etc.), finding strict bounds on the
limits of roots, and unifying a number of results in the literature. Second, we
pose a stable numerical method which is guaranteed to quickly and robustly
converge to the separatrix. This new approach is implemented in the Black Hole
Perturbation Toolkit, and results in a ~45x speedup over the prior robust
approach.
