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  "slug": "affine-elementary-symmetric-total-nonnegativity",
  "title": "Coefficientwise total nonnegativity for affine elementary-symmetric sequences",
  "shortTitle": "One affine factor, every minor",
  "url": "https://evidencepress.org/releases/affine-elementary-symmetric-total-nonnegativity/",
  "oneLine": "A uniform proof candidate gives nonnegative monomial coefficients for every Toeplitz minor of the single affine elementary-symmetric sequence.",
  "abstract": "For b_k=(1+kt)e_k(X), this unrefereed candidate supplies a uniform written proof of coefficientwise total nonnegativity for all alphabet sizes and finite Toeplitz minor orders, including arbitrary index gaps. A diagonal rank-one realization, repeated-index immanants and orthogonal tensor grading yield individual nonnegative coefficients. Skew Schur expansion and polynomial homogenization extend the conclusion to (uk+v)e_k(X), including v=0. Multiple affine factors and general Hadamard conjectures are excluded. Classical antecedents are credited; historical priority remains uncertain.",
  "datePublished": "2026-09-07",
  "dateModified": "2026-09-07",
  "version": "0.1.0-candidate",
  "doi": "10.5281/zenodo.22649676",
  "doiUrl": "https://doi.org/10.5281/zenodo.22649676",
  "conceptDoi": "10.5281/zenodo.22649675",
  "pdfUrl": "https://github.com/ipitchford/affine-elementary-symmetric-total-nonnegativity/releases/download/v0.1.0-candidate/paper.pdf",
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  "authors": [
    "Anonymous"
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  "status": "unrefereed-candidate",
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      "note": "Fresh extraction; 4725 coefficient comparisons and scoped controls. Not a formal proof.",
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    "name": "AIM total-positivity workshop: Monomial-TNN Toeplitz matrices, affine branch",
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  "keywords": [
    "total nonnegativity",
    "Toeplitz minors",
    "symmetric functions",
    "immanants",
    "affine multipliers"
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  "keyResults": [
    "All-alphabet and all-order coefficientwise total nonnegativity for one normalized affine multiplier.",
    "Two-parameter homogenization including v=0.",
    "Explicit projected-norm coefficient formula and arbitrary-gap skew-shape bridge.",
    "Order-three counterexample to coefficientwise Schur positivity."
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    {
      "citation": "AIM (2023), Theory and applications of total positivity, report p.2: affine monomial-TNN question.",
      "url": "https://aimath.org/pastworkshops/totalposrep.pdf",
      "doi": null
    },
    {
      "citation": "Sokal (2025), positivity conjectures, slides 11–12: affine-product family and Stanley strategy.",
      "url": "https://sites.math.rutgers.edu/~zeilberg/expmath/sokal2025.pdf",
      "doi": null
    },
    {
      "citation": "Carvalho and Freitas (2014), character-symmetric power derivatives: positive-direction tensor antecedent.",
      "url": "https://doi.org/10.13001/1081-3810.1620",
      "doi": "10.13001/1081-3810.1620"
    },
    {
      "citation": "Huber and Maassen (2021), Matrix forms of immanant inequalities, section 2.",
      "url": "https://arxiv.org/abs/2103.04317",
      "doi": "10.48550/arXiv.2103.04317"
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