{
  "abstract": "We classify the nonzero complete Bernstein functions whose matrix functions preserve every real P-matrix in every order. They are precisely $f(z)=a+bz$, with $a,b\\ge0$ and $a+b>0$, and $f(z)=a+cz/(z+s)$, with $a\\ge0$ and $c,s>0$. For each fixed order $n\\ge4$, the preservers vanishing at zero are exactly $cz$ and $cz/(z+s)$; every other such function admits a strictly totally positive counterexample. For a finite positive measure $μ$ on $(0,\\infty)$, put $g(z)=\\int z/(z+s)\\,μ(ds)$ and $M_t=\\int s^t\\,μ(ds)$. In every fixed order $n\\ge4$, a finite constant shift restores preservation on all real P-matrices if and only if $M_{r_n}$ and $M_{-r_n}$ are finite, where $r_n=\\lfloor n/2\\rfloor-1$. Every $a>c_n\\sqrt{M_{r_n}M_{-r_n}}$ is sufficient, with $c_n$ depending only on $n$. A complete Bernstein function vanishing at zero admits a possibly different repairing constant in every order exactly when it is linear, or is a pure kernel mixture with all positive and negative integer moments finite. For principal real powers, the preserving exponents are $\\mathbb R$ in order one, $[-1,1]$ in orders two and three, and $\\{-1,0,1\\}$ in every higher order. An explicit rational determinant-one totally positive family gives negative principal minors uniformly on prescribed compact subsets of $(-1,0)\\cup(0,1)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.00096",
  "author_total": 1,
  "authors": [
    {
      "name": "Wei Xie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1r98chnww1ks6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.00096",
  "fulltext_url": null,
  "id": "1r98chnww1ks6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.00096",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.RA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.00096",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1r98chnww1ks6.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.00096",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.00096"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Complete Bernstein functions and real powers of P-matrices",
  "title_text": "Complete Bernstein functions and real powers of P-matrices",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/1r98chnww1ks6",
  "venue": "arXiv",
  "versions": []
}