{
  "schema_version": 1,
  "problem_number": "AIM-ALGEBRAIC_GEOMETRY-0185",
  "title": "Base-Divisor Degree Formulas for Two-View Algebraic Curves",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let two fixed projective cameras with distinct centers have integral plane image curves of degrees m and n, neither an epipolar line, over an algebraically closed field of characteristic zero. We describe all integral reconstructions using the normalized components of the fiber product of the moving epipolar pencils. If a component has pencil degree h and pulled-back epipolar base divisors B_X and B_Y, its triangulated world curve has degree h + deg max(B_X, B_Y). This retains singular branches and images passing through the epipoles. We derive the total residual degree and an explicit formula for the baseline cycle multiplicity in terms of branch base orders and pencil ramification indices. When both moving pencils have degree one, the compatible integral world curve is unique, and the possible degrees for fixed m and n are exactly max(m,n), ..., m+n-1. The classical fiber-product method and twisted-cubic example are credited; the formulas here make their base-divisor corrections explicit. This is a complete scoped theorem related to AIM's fixed-camera compatibility question, not a claim to resolve every interpretation of that question. The preprint is unrefereed and makes no absolute priority claim.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.AG"
  ],
  "keywords": [
    "AIM-ALGEBRAIC_GEOMETRY-0185",
    "algebraic vision",
    "epipolar geometry",
    "base divisors",
    "fiber product",
    "reconstruction degree"
  ],
  "manuscript_version_date": "2026-10-23",
  "publication_date": "2026-10-23",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-23",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-algebraic-geometry-0185/",
  "pdf_url": "https://eulersolve.org/papers/aim-algebraic-geometry-0185/paper.pdf?v=cce0e0cc5919",
  "doi": "10.5281/zenodo.23604002",
  "zenodo_record_url": "https://zenodo.org/records/23604002",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Over an algebraically closed characteristic-zero field, for distinct-center cameras and integral non-epipolar-line images, normalized moving-pencil fiber-product components exhaust all integral reconstructions and have degree h + deg max(B_X,B_Y). The branch formula gives the baseline cycle coefficient; degree-one pencils have the sharp spectrum max(m,n) through m+n-1. The open-ended AIM compatibility question remains partially solved. Real visibility, general smoothness, nonreduced inputs and the full baseline primary ideal are outside this theorem. Classical methods and the twisted-cubic example are credited; precise novelty is not certified. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
  "files": {
    "paper.pdf": {
      "sha256": "cce0e0cc5919b0490f521f86b11865ccd9625c02451c0a007f86d2b06710c9ac"
    },
    "source.zip": {
      "sha256": "7f2971b7b12d75cc9df0889c001dd867f7b04e9780d2daa725b0f87e86c1780a"
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    "verification_report.md": {
      "sha256": "b3adf8e4ded0191888da02ae15f934982f31f6c922fb60a365ee4c89f2c6c34c"
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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."
}