{
  "abstract": "Fix $I = (0,ρ)$, where $0<ρ\\leq\\infty$, and let $\\mathbb{P}_n(I)$ be the set of positive semidefinite $n\\times n$ matrices with entries in $I$. A longstanding problem in matrix theory is to characterize the functions $f: I \\to \\mathbb{R}$ for which the entrywise calculus $f[A] = [f(a_{ij})]_{i,j = 1}^{n}$ preserves positive semidefiniteness for all $A \\in \\mathbb{P}_n(I)$. We characterize these functions exactly: if $f\\in C^{2n-2}(I)$ and $\\mathcal{E} = x\\frac{d}{dx}$, then this holds if and only if $$ f^{(k)}(x)\\geq 0 \\quad (0\\leq k\\leq n-1) \\qquad\\text{and}\\qquad \\bigl[\\mathcal{E}^{i+j}f(x)\\bigr]_{i,j = 0}^{n-1}\\succeq 0 $$ for every $x\\in I$. Regularization then removes all a priori smoothness: for $n\\geq2$, every preserver belongs to $C^{2n-4}(I)$ and the same characterization holds by interpreting the last two derivatives in the sense of distributions. As applications, we recover classical results of FitzGerald--Horn and Vasudeva, and obtain a complete classification of generalized polynomials with prescribed real exponents and arbitrary coefficients. We also determine optimal constants in entrywise domination inequalities under finite regularity, extend the sharp finite-sum thresholds of Belton--Guillot--Khare--Putinar and Khare--Tao to positive mixtures of powers, and answer a question of Khare and Tao by showing that no finite collection of matrices with entries strictly inside $I$ can detect positivity preservation on $\\mathbb{P}_n(I)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.15904",
  "author_total": 2,
  "authors": [
    {
      "name": "Ludovick Bouthat"
    },
    {
      "name": "Dominique Guillot"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fbtb68pbvc846.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.15904",
  "fulltext_url": null,
  "id": "fbtb68pbvc846",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.15904",
  "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.CA"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Mathematical Inequalities and Applications"
    ]
  ],
  "openalex": "W7203688590",
  "pdf_url": "https://arxiv.org/pdf/2608.15904",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fbtb68pbvc846.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:01Z",
      "native_id": "2608.15904",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.15904"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A Finite-order Characterization of Entrywise Positivity Preservers",
  "title_text": "A Finite-order Characterization of Entrywise Positivity Preservers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:01Z",
  "url": "https://research.draconitic.com/paper/fbtb68pbvc846",
  "venue": "arXiv",
  "versions": []
}