Hilbert's sixteenth problem: uniform bounds for limit cycles

What OpenAI claims

Resolves the uniform boundedness assertion in Hilbert's sixteenth problem: the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on its degree. For classical quintic Liénard systems, the exact maximum is two limit cycles. (Lean)

Papers (2)

1. Uniform bounds for planar polynomial limit cycles

   24 September 2026 · Preprint, not peer reviewed · PDF

   For every degree, we prove that the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on that degree. This establishes the uniform boundedness assertion in the second part of Hilbert's sixteenth problem. The proof uses separation of asymptotic expansions on nested complex domains and a finite-dimensional counting argument.

2. Two limit cycles for quintic Liénard systems

   24 September 2026 · Preprint, not peer reviewed · PDF

   Every classical Liénard system ẋ=y-F(x), ẏ=-x, with F an arbitrary real polynomial of degree at most five, has at most two geometrically distinct isolated periodic orbits, and the bound is attained. This proves the degree-five case of the Lins Neto–de Melo–Pugh conjecture.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.