{
  "id": "2308.14722",
  "version": "v1",
  "published": "2023-08-28T17:24:49.000Z",
  "updated": "2023-08-28T17:24:49.000Z",
  "title": "higher derivatives of functions with given critical points and values",
  "authors": [
    "Gil Goldman",
    "Yosef Yomdin"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "Let $f: B^n \\rightarrow {\\mathbb R}$ be a $d+1$ times continuously differentiable function on the unit ball $B^n$, with $\\max_{z\\in B^n} \\|f(z)\\|=1$. A well-known fact is that if $f$ vanishes on a set $Z\\subset B^n$ with a non-empty interior, then for each $k=1,\\ldots,d+1$ the norm of the $k$-th derivative $\\|f^{(k)}\\|$ is at least $M=M(n,k)>0$. A natural question to ask is ``what happens for other sets $Z$?''. This question was partially answered in [16]-[18]. In the present paper we ask for a similar (and closely related) question: what happens with the high-order derivatives of $f$, if its gradient vanishes on a given set $\\Sigma$? And what conclusions for the high-order derivatives of $f$ can be obtained from the analysis of the metric geometry of the ``critical values set'' $f(\\Sigma)$? In the present paper we provide some initial answers to these questions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-28T17:24:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "higher derivatives",
      "critical points",
      "high-order derivatives",
      "well-known fact",
      "natural question"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}