  When geodesic equations are formulated in terms of an effective potential
$U$, circular orbits are characterised by $U=\partial_a U=0$. In this paper we
consider the case where $U$ is an algebraic function. Then the condition for
circular orbits defines an $A$-discriminantal variety. A theorem by Rojas and
Rusek, suitably interpreted in the context of effective potentials, gives a
precise criteria for certain types of spacetimes to contain at most two
branches of light rings (null circular orbits), where one is stable and the
other one unstable. We identify a few classes of static, spherically-symmetric
spacetimes for which these two branches occur and show that the spacetimes with
non-degenerate horizons do not have stable light rings.
