{
  "id": "2411.09594",
  "version": "v1",
  "published": "2024-11-14T17:06:06.000Z",
  "updated": "2024-11-14T17:06:06.000Z",
  "title": "A note on a recent attempt to solve the second part of Hilbert's 16th Problem",
  "authors": [
    "Claudio A. Buzzi",
    "Douglas D. Novaes"
  ],
  "categories": [
    "math.DS"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-14T17:06:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34C07",
      "34C23",
      "37G15"
    ],
    "keywords": [
      "second part",
      "limit cycle",
      "hilberts 16th problem asks",
      "planar polynomial vector fields",
      "maximum number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}