{
  "abstract": "For an $n\\times n$ oscillatory matrix $A$, let $e_k(A)$ be the least positive integer $m$ for which every minor of order $k$ of $A^m$ is positive. We determine the profile $(e_1(A),\\ldots,e_n(A))$. For every nonsingular totally nonnegative matrix, the column index sets of positive entries in each compound row form an interval in the componentwise order. The endpoint maps are order-preserving and compose under multiplication. This yields a two-corner criterion for positivity of all minors of a fixed order. Consequently, $e_k(A)$ is the larger of the first positivity times of two remote corner minors. We prove the sharp inequalities $e_k(A)\\leq\\max{k,n-k}$ for $1\\leq k<n$ and $|e_{k+1}(A)-e_k(A)|\\leq1$ for $1\\leq k\\leq n-2$. For $D=\\operatorname{diag}(1,-1,1,-1,\\ldots)$, the matrix $DA^{-1}D$ is oscillatory and satisfies $e_k(DA^{-1}D)=e_{n-k}(A)$ for $1\\leq k<n$; the determinant exponent remains one. The ordered positions of the positive factors in an adjacent bidiagonal factorization determine the profile independently of their values. Each one-sided factor sequence can be represented by a single permutation, giving a finite characterization of all profiles and an integer unimodular realization of each. We also obtain a compatibility condition across minor orders: if $3\\leq k\\leq n-3$ and $e_k(A)\\leq2$, then $e_j(A)\\leq2+\\lceil |j-k|/2\\rceil$ for $2\\leq j\\leq n-2$. In particular, $e_k(A)\\leq2$ implies $e_{k+2}(A)\\leq3$ for $3\\leq k\\leq n-4$, and both the constant and this range are sharp.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05787",
  "author_total": 1,
  "authors": [
    {
      "name": "Wei Xie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f7jem8drzd7zm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05787",
  "fulltext_url": null,
  "id": "f7jem8drzd7zm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05787",
  "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.CO"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "1703|Matrix Theory and Algorithms"
    ]
  ],
  "openalex": "W7212015720",
  "pdf_url": "https://arxiv.org/pdf/2609.05787",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f7jem8drzd7zm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.05787",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05787"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Minor-Order Exponent Profiles of Oscillatory Matrices",
  "title_text": "Minor-Order Exponent Profiles of Oscillatory Matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/f7jem8drzd7zm",
  "venue": "arXiv",
  "versions": []
}