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  "slug": "sharp-local-minor-ratios",
  "title": "Sharp local minor-ratio conditions for total positivity",
  "shortTitle": "A sharp threshold for positive minors",
  "url": "https://evidencepress.org/releases/sharp-local-minor-ratios/",
  "oneLine": "A proof candidate finds the exact strength of local minor ratios needed to force total positivity.",
  "abstract": "For a real n-square matrix and 1≤k<n, assume every minor through order k is positive. On every (k+1)-square submatrix, use its first k−1 rows and columns as a common pivot and compare the two products of bordered k-minors. Requiring their ratio to exceed 4 cos²(π/(n−k+2)) forces total positivity, and every smaller threshold admits a matrix with all proper minors positive but negative determinant. The optimal dimension-independent threshold is 4. This is an unrefereed written proof candidate for this precise condition, not a claim of identity with an omitted AIM workshop formula.",
  "datePublished": "2026-09-07",
  "dateModified": "2026-09-07",
  "version": "0.1.0-candidate",
  "doi": "10.5281/zenodo.22650759",
  "doiUrl": "https://doi.org/10.5281/zenodo.22650759",
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  "pdfUrl": "https://github.com/ipitchford/sharp-local-minor-ratios/releases/download/v0.1.0-candidate/sharp-local-minor-ratios-0.1.0-candidate.pdf",
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  "repoUrl": "https://github.com/ipitchford/sharp-local-minor-ratios",
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    "Anonymous"
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      "note": "Ten rational examples, 53 symbolic identities, review regression and five rejection controls. Finite corroboration only.",
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  "problem": {
    "name": "AIM-LINEAR_ALGEBRA-0017 / AIM Problem 1.65",
    "url": "http://aimpl.org/totalpos/1/"
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  "keywords": [
    "total positivity",
    "minor ratios",
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  "keyResults": [
    "Exact dimension-dependent threshold for the specified common-pivot local condition.",
    "Optimal dimension-independent threshold 4 for each fixed minor order.",
    "Scale-separated bordering transfers all local ratios simultaneously.",
    "Explicit nonvacuity and a relative minor-data perturbation bound."
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    {
      "citation": "Johnson and Karp, AIM Problem 1.65.",
      "url": "http://aimpl.org/totalpos/1/",
      "doi": null
    },
    {
      "citation": "Katkova and Vishnyakova, consulted preprint Theorem 1(ii).",
      "url": "https://arxiv.org/abs/math/0504183v1",
      "doi": null
    },
    {
      "citation": "Fallat, Johnson and Sokal (2017), with 2021 corrigendum; exterior bordering comparison.",
      "url": "https://arxiv.org/abs/1612.02210v2",
      "doi": "10.48550/arXiv.1612.02210"
    },
    {
      "citation": "Gasca and Mühlbach (1987), generalized Schur-complement tests; abstract only inspected.",
      "url": "https://doi.org/10.1016/0168-9274(87)90049-3",
      "doi": "10.1016/0168-9274(87)90049-3"
    },
    {
      "citation": "Fallat, Gupta and Johnson, Sufficient conditions for total positivity, compounds, and Dodgson condensation.",
      "url": "https://arxiv.org/abs/2405.06069v1",
      "doi": "10.48550/arXiv.2405.06069"
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