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.

翻译：在本文中, 我们考虑确定 多面平流的周期轨道数的问题。 这个问题是Hilbert第16个问题的变体。 我们使用对可计算性的自然定义, 表明这一问题一方面不可计算, 另一方面, 统一地计算在单位磁盘上定义的所有结构稳定的系统上。 我们还证明, 有一组多面平流系统, 对其周期轨道数没有可比较的尖锐上限 。