{
  "schema_version": 1,
  "problem_number": "AMR-029-0087",
  "title": "Divisorial Gonality of Banana Trees in Quadratic Time",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "A banana tree is a connected loopless multigraph whose underlying simple graph is a tree. We give an exact algorithm that computes its divisorial gonality and an effective divisor attaining it in O(n^2) word operations and O(n^2) space after reading the input, where n is the number of vertices. Multiplicities may be supplied in binary. A rooted residue normal form reduces positive rank to positivity of one pile at every root. A dynamic program combines independent child costs conditional on the parent pile, and sliding-window minima evaluate the recurrence in quadratic time. The result covers arbitrary branching, extending the cubic path algorithm of Beougher et al. The gonality conjecture for banana trees was already proved in that work; this paper does not resolve the general multigraph conjecture. The proof is self-audited, AI-assisted and unrefereed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO",
    "cs.DS"
  ],
  "keywords": [
    "divisorial gonality",
    "chip firing",
    "banana tree",
    "dynamic programming",
    "sliding-window minimum",
    "graph divisor",
    "AMR-029-0087"
  ],
  "manuscript_version_date": "2026-10-19",
  "publication_date": "2026-10-19",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-19",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-029-0087/",
  "pdf_url": "https://eulersolve.org/papers/amr-029-0087/paper.pdf?v=b98b088078e2",
  "doi": "10.5281/zenodo.23467234",
  "zenodo_record_url": "https://zenodo.org/records/23467234",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete proof of the quadratic exact gonality algorithm and divisor reconstruction for arbitrary banana trees. The complexity is a word-operation bound after reading binary multiplicities, with division on O(log n)-bit words. The general multigraph genus bound in AMR-029-0087 remains unresolved; no whole-record closure is claimed. Beougher et al. already proved the genus bound for banana trees and a cubic algorithm for banana paths, and are credited. Finite exact checks support but do not replace the proof. AI-assisted, self-audited and unrefereed; no independent review, formal certification, dataset-maintainer acceptance or absolute priority is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "b98b088078e2d938171463c350bb5914338e8831246cff942eabceeb61c78002"
    },
    "source.zip": {
      "sha256": "1559cc7087a1121f55563a20e2b9c53b6b13cfeefd3edda25ad25047bc69613b"
    },
    "verification_report.md": {
      "sha256": "6c5785cabbac63e312b6a2601bf2451f7b2d24c1a152c1f76fbc2a968d425f88"
    }
  },
  "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."
}