Minimum dimension for undecidable homogeneous cubic stability

Determine the least dimension in which global asymptotic stability or Lyapunov stability of rational homogeneous cubic vector fields is undecidable, including whether either problem is undecidable in dimension three.

Background

The paper proves that global asymptotic stability is many-one complete for homogeneous cubic vector fields in every dimension above a sufficiently large threshold, but the threshold dimension is not computed and is explicitly described as very large. It also establishes analogous hardness results for Lyapunov stability in sufficiently large dimensions. The authors therefore leave unresolved the smallest dimension in which either stability problem becomes undecidable, with dimension three identified as a particularly important case because Arnold suggested it as a possible threshold.