{
  "schema_version": 1,
  "problem_number": "AMR-152-0049",
  "title": "Quadratic Hénon Maps over Q with Rational Cycles of Length Three and Four for Every Jacobian",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "In a list of open problems on Hénon maps (Arnold Math. J. 10 (2024)), P. Ingram conjectured that for all but finitely many δ ∈ Q every Q-rational periodic point of every map f(x, y) = (y, y² + c − δx) with c ∈ Q has period dividing 2. We show that the statement, as printed, is false for every δ ∈ Q, in each reading the text supports (finitely many exceptional δ; for fixed δ, finitely many exceptional c; finitely many exceptional pairs). For δ ≠ 1 and c = −(29δ² − 38δ + 29)/16 the point ((5 − 7δ)/4, −(1 + δ)/4) has period 3; for δ = 1 and c = −1 − s², s ∈ Q, there is a rational 3-cycle; for every δ ≠ 0 an explicit c gives a rational 4-cycle. Such cycles are essentially implicit in the source: the interpolation construction that opens the same section yields rational 4-cycles for every δ ∉ {0, 1}, and our 4-cycles are its normal form; so the printed conjecture contradicts its own section and is probably not the intended statement. We also describe all rational 3-cycles: shifted by 1/2, their x-coordinates are the rational points, off the line u = v = w, of the cone over the nodal cubic u³ + v³ + w³ − u²v − v²w − w²u = 0. Hence every δ ≠ 1 has infinitely many c with a rational 3-cycle, and for δ = 1 one exists if and only if −(1 + c) is a rational square. Computations, which are not theorems, find for δ = −1 exactly Ingram's periods 1, 2, 3, 4, 6, 8, and a rational 5-cycle for δ = −4 and, by a duality δ ↔ 1/δ, for δ = −1/4. The case δ = 0 is the classical map z² + c (Morton; Morton and Raianu), the case δ = −1 is Ingram's, and the novelty of the note is modest. This is an unrefereed note.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "keywords": [
    "quadratic Hénon maps",
    "rational periodic points",
    "rational cycles",
    "arithmetic dynamics",
    "Jacobian",
    "nodal cubic",
    "dynatomic curves",
    "uniform boundedness",
    "disproof",
    "Hénon maps: a list of open problems",
    "Arnold Mathematical Journal problems",
    "UnsolvedMath",
    "AMR-152-0049",
    "math.DS",
    "math.NT",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-18",
  "publication_date": "2026-10-18",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-18",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-152-0049/",
  "pdf_url": "https://eulersolve.org/papers/amr-152-0049/paper.pdf?v=74135c95126b",
  "doi": "10.5281/zenodo.23445974",
  "zenodo_record_url": "https://zenodo.org/records/23445974",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The printed conjecture contradicts the interpolation construction that opens its own section and is probably misstated; the note gives explicit families of rational 3- and 4-cycles and describes all rational 3-cycles. Novelty modest. Unrefereed note.",
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    "source.zip": {
      "sha256": "24629d3897d95a64faa96f8a09522af114cb714b1f2d88c057540bfc61028615"
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    "verification_report.md": {
      "sha256": "07b3ac41a450087cb08894b3a228fabdd730e112de05b8610f17614f8eb6bc07"
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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."
}