{
  "schemaVersion": "1.1",
  "slug": "degree-difference-affine-slices",
  "title": "The degree-difference principle and affine slices of binary-form factorisation spaces",
  "shortTitle": "Degree-difference principle and affine slices",
  "url": "https://evidence-press.pages.dev/releases/degree-difference-affine-slices/",
  "oneLine": "A clean determinant identity for the map that multiplies two polynomials and records their resultant — and a study of the geometric slices where Jacobian-conjecture-style questions live.",
  "abstract": "Multiply two polynomials together and, alongside the product, record their resultant — a single number measuring whether they share a root. This paper proves an exact formula for the Jacobian determinant of that combined operation: up to sign, it is the degree difference of the two polynomials times the square of their resultant. From this 'degree-difference principle' the paper develops the geometry of natural affine slices of binary-form factorisation spaces — the setting in which the project's related Jacobian-conjecture investigations take place — including classifications of low-degree slices, explicit coordinates for Keller maps, and Euler-characteristic obstructions in higher degrees.",
  "datePublished": "2026-07-28",
  "dateModified": "2026-07-28",
  "version": "0.1-candidate",
  "doi": "10.5281/zenodo.21647593",
  "doiUrl": "https://doi.org/10.5281/zenodo.21647593",
  "conceptDoi": "10.5281/zenodo.21647592",
  "pdfUrl": "https://github.com/ipitchford/degree-difference-affine-slices/releases/download/v0.1-candidate/paper.pdf",
  "altPdfUrl": "https://raw.githubusercontent.com/ipitchford/degree-difference-affine-slices/main/paper.pdf",
  "zenodoUrl": "https://zenodo.org/records/21647593",
  "repoUrl": "https://github.com/ipitchford/degree-difference-affine-slices",
  "releaseUrl": "https://github.com/ipitchford/degree-difference-affine-slices/releases",
  "markdownUrl": "https://evidence-press.pages.dev/releases/degree-difference-affine-slices/index.md",
  "bibtexUrl": "https://evidence-press.pages.dev/releases/degree-difference-affine-slices/cite.bib",
  "audioUrl": "https://evidence-press.pages.dev/assets/audio/degree-difference-affine-slices.mp3",
  "imageUrl": "https://evidence-press.pages.dev/assets/og/degree-difference-affine-slices.png",
  "coverArtUrl": "https://evidence-press.pages.dev/assets/art/degree-difference-affine-slices.svg",
  "media": [],
  "authors": [
    "OpenAI Codex / Anthropic models"
  ],
  "license": "CC0-1.0",
  "status": "unrefereed-candidate",
  "verification": {
    "peerReviewed": false,
    "independentlyReproduced": false,
    "formallyVerified": false,
    "internallyReplayed": true,
    "detail": "Unrefereed candidate manuscript. No peer review, no independent reproduction, and no documented external human review. Computational verification covers only the explicit low-degree polynomial identities (bidegrees r+s ≤ 4; sign checks to degree eight); the general all-degree determinant argument and the theoretical proofs (torsor, divisor-class, cubic-orbit classification) are asserted but not machine-checked. The manuscript is AI-attributed with human direction."
  },
  "provenance": {
    "aiGenerated": true,
    "generatedBy": [
      "OpenAI Codex / Anthropic models"
    ],
    "humanRole": "problem selection, mediation, and publication management",
    "disclosure": "The mathematics/research in this release was generated by AI systems as credited; see the Zenodo record for full attribution."
  },
  "problem": {
    "name": "Structure of binary-form factorisation spaces (adjacent to the Jacobian conjecture)",
    "url": "https://en.wikipedia.org/wiki/Jacobian_conjecture"
  },
  "keywords": [
    "algebraic geometry",
    "binary forms",
    "resultant",
    "Sylvester matrix",
    "Jacobian conjecture",
    "Keller maps",
    "affine slices",
    "factorisation spaces",
    "computer-assisted mathematics",
    "symbolic verification",
    "SymPy"
  ],
  "keyResults": [
    "Determinant identity: det DΦ_{r,s} = (−1)^{s(r+1)} (r−s) · Res(A,B)² for the multiplication–resultant map Φ_{r,s}(A,B) = (AB, Res(A,B)).",
    "Associated torsor and divisor-class statements for the factorisation spaces.",
    "Classification of normalised linear–quadratic slices.",
    "Explicit coordinates for tangent nonosculating slices and the induced Keller maps.",
    "Fibre and image characterisation of the map.",
    "Euler-characteristic obstructions for higher-degree cases."
  ],
  "evidencePackage": "Manuscript (PDF and TeX); a SymPy verification script with intentional negative-control tests, confirming the polynomial identities through bidegree r+s ≤ 4 and base-point signs through degree eight; a claim-level evidence map (AI_INDEX.md); assurance-boundary documentation (STATUS.md, ASSURANCE.md); provenance records; SHA-256 manifests; Zenodo and Software Heritage archives.",
  "openProblems": [
    "Verify the all-degree determinant identity by an independent conceptual proof — or locate it in prior literature on resultants and multiplication maps, where such identities may already exist in another form.",
    "Subject the torsor and divisor-class arguments, which the symbolic checker deliberately does not cover, to expert human review.",
    "Extend the symbolic verification beyond bidegree 4 and automate the identity checks at general degree.",
    "Determine whether the Euler-characteristic obstructions can be upgraded to full non-isomorphism proofs in the higher-degree cases.",
    "Connect the classified slices explicitly to the companion releases on exotic affine spheres and reducible incidence divisors, which build on this framework."
  ],
  "relatedWorks": [
    {
      "citation": "Keller, O.-H. (1939). Ganze Cremona-Transformationen — origin of the Jacobian conjecture, which remains open even for two variables.",
      "url": "https://en.wikipedia.org/wiki/Jacobian_conjecture"
    },
    {
      "citation": "Companion release: Exotic affine three-spheres and the quadratic–cubic obstruction.",
      "url": "https://doi.org/10.5281/zenodo.21647607"
    },
    {
      "citation": "Companion release: Reducible incidence divisors and the isolation of affine slices.",
      "url": "https://doi.org/10.5281/zenodo.21647616"
    }
  ]
}