  In this paper, we consider the problem of determining the \emph{exact} number
of periodic orbits for polynomial planar flows. This problem is a variant of
Hilbert's 16th problem. Using a natural definition of computability, we show
that the problem is noncomputable on the one hand and, on the other hand,
computable uniformly on the set of all structurally stable systems defined on
the unit disk. We also prove that there is a family of polynomial planar
systems which does not have a computable sharp upper bound on the number of its
periodic orbits.
