This paper proves a clear but surprising fact: there are some dimensions N and odd degrees D for which no computer program can take the rational coefficients of a homogeneous polynomial vector field and always decide whether the origin is Lyapunov stable for the ordinary differential equation (ODE) ˙Y = F(Y). Lyapunov stability means that every solution that starts close to the origin stays close forever. “Undecidable” means there is no algorithm that gives the correct yes/no answer for every such input field.

The authors give a concrete kind of proof called a reduction from a known undecidable problem. They start from a fixed register machine (a simple abstract computer) whose halting problem is undecidable. From the code of that machine and an input number, they build, by a sequence of explicit steps, a homogeneous polynomial vector field with rational coefficients. The construction is algorithmic: given the machine and input, one can write down the field. They then show that the origin of the resulting ODE is Lyapunov stable exactly when the machine halts on that input. Because the machine-halting question has no general algorithm, neither does the stability question for these polynomial fields.

A few technical ideas make the reduction work. Homogeneity of the field (every term has the same degree) turns the local idea “stay close” into a global, uniform growth bound. Using a polar-like change of variables, the authors isolate a radial rate: how solutions grow or shrink in radius while their direction moves on the unit sphere. They package this into an “accumulated output,” which is the integral over time of the radial rate along sphere trajectories. Then they build a projective lift: a homogeneous polynomial field whose stability is equivalent to the boundedness of that accumulated output. The chain of constructions also includes devices to handle arbitrary initial states, a clocked relaxation mechanism, smoothing of non-polynomial pieces, and a way to encode the register machine’s behavior.