{
  "schema_version": 1,
  "problem_number": "AIM-LINEAR_ALGEBRA-0012",
  "title": "Weighted Root Deletions and Coefficientwise Toeplitz Positivity",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For independent indeterminates a, x_1,...,x_N, y_1,...,y_N, we prove that the sequence b_k = a e_k(X) + sum_i y_i e_k(X without x_i) is coefficientwise totally nonnegative: every minor of its upper Toeplitz matrix has nonnegative integer coefficients. In particular, this holds for the coefficients of (u D_z + v) product_i(1+x_i z), coefficientwise in u,v,X. The proof realizes the sequence as sums of bordered principal minors of a star-shaped Gram pencil. A maximal-weight compression in a Schur module expresses the necessary Schur-complement characters as traces against orthogonal projections. An additional letter-content grading separates the independent root weights and yields explicit squared-norm coefficient certificates. This resolves the derivative-plus-constant branch of an AIM total-positivity question, not its other operator conjectures. The note is unrefereed and makes no absolute priority claim.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "math.CA"
  ],
  "keywords": [
    "coefficientwise total positivity",
    "Toeplitz minors",
    "elementary symmetric functions",
    "polynomial derivatives",
    "Schur complements",
    "Schur modules",
    "Gram matrices",
    "weighted root deletion",
    "AIM-LINEAR_ALGEBRA-0012",
    "math.CO",
    "math.CA"
  ],
  "manuscript_version_date": "2026-09-28",
  "publication_date": "2026-09-28",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-28",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-linear-algebra-0012/",
  "pdf_url": "https://eulersolve.org/papers/aim-linear-algebra-0012/paper.pdf?v=376539b56e72",
  "doi": "10.5281/zenodo.23012273",
  "zenodo_record_url": "https://zenodo.org/records/23012273",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "This is a complete independent-weight root-deletion theorem and a solution of the D+a (equivalently uD+v) branch, not a solution of every operator conjecture in AIM Problem 1.4. The prior anonymous affine theorem and classical methods are credited in the manuscript. No independent peer review, formal verification or absolute priority certificate is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "376539b56e7250b38dcde50558fb3a5617baf20c616b226d65da7d9bb14d4e2d"
    },
    "source.zip": {
      "sha256": "7883db591430c7ee074b6a0a35ee9844e73ba9ea2d2aafe09a7dd607a130ad73"
    },
    "verification_report.md": {
      "sha256": "904e6f6706287a2a2896058cca855eb525db78c7c993da2974b45a45bf4c3e2c"
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  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}