A note on a recent attempt to solve the second part of Hilbert's 16th Problem

TL;DR

This paper critically examines a recent claim of solving Hilbert's 16th Problem, highlighting contradictions with known asymptotic growth of the maximum number of limit cycles in polynomial vector fields.

Contribution

The authors analyze and challenge a recent proposed solution to Hilbert's 16th Problem, providing counterexamples and discussing inconsistencies with established growth estimates.

Findings

The proposed solution contradicts known asymptotic growth of limit cycles.

Counterexamples demonstrate the proposed formula's inaccuracies.

The paper clarifies the limitations of recent claims on Hilbert's 16th Problem.

Abstract

For a given natural number $n$, the second part of Hilbert's 16th Problem asks whether there exists a finite upper bound for the maximum number of limit cycles that planar polynomial vector fields of degree $n$ can have. This maximum number of limit cycle, denoted by $H(n)$, is called the $n$th Hilbert number. It is well-established that $H(n)$ grows asymptotically as fast as $n^2 \log n$. A direct consequence of this growth estimation is that $H(n)$ cannot be bounded from above by any quadratic polynomial function of $n$. Recently, the authors of the paper [Exploring limit cycles of differential equations through information geometry unveils the solution to Hilbert's 16th problem. Entropy, 26(9), 2024] affirmed to have solved the second part of Hilbert's 16th Problem by claiming that $H(n) = 2(n - 1)(4(n - 1) - 2)$. Since this expression is quadratic in $n$, it contradicts the established asymptotic behavior and, therefore, cannot hold. In this note, we further explore this issue by discussing some counterexamples.