{
  "schema_version": 1,
  "problem_number": "AMR-042-0001",
  "title": "The Exact Order of Mixing in a Problem of Ward: A Seven-Term Polynomial with No Sparser Multiple",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Problem A of T. Ward's list \"Six problems in algebraic dynamics\" concerns the polynomial f = 1 + u_1 u_2 + u_1^2 u_2 + u_1^3 u_2 + u_1^4 + u_2^2 + u_1^4 u_2^2 and the algebraic Z^2-action attached to the module Z[u_1^{±1}, u_2^{±1}]/⟨p, f⟩, where p is a prime for which f is irreducible modulo p. Its order of mixing M was known to satisfy 3 ≤ M < 7, and the problem asks for the exact value. We show that M = 6 for every prime p: the action is mixing of order 6 and not of order 7. The polynomial f is absolutely irreducible modulo every prime, so no prime has to be excluded. By results of Schmidt, Masser, and Derksen and Masser, and because the radical of the group generated by u_1, u_2 in the function field of the curve f = 0 consists only of constant multiples of monomials, M + 1 is the least number of terms of a non-zero multiple of f in F_p[u_1^{±1}, u_2^{±1}]. Our main result is that every non-zero multiple of f with coefficients in an algebraic closure of F_p has at least seven terms. The proof is elementary: it sorts a multiple by powers of u_2, reduces to two possible shapes of a six-term multiple by a Chebyshev-type recurrence, and excludes these by a local analysis at the points of the curve with u_1^4 = −1, a Frobenius splitting and an explicit computation. The prime 2 is treated separately. For p = 2 an exhaustive computer search confirms the main result for coefficients in F_2, independently of this proof, and shows that the support of f is, up to equivalence, the only non-mixing set with seven points. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.DS",
    "math.NT"
  ],
  "keywords": [
    "order of mixing",
    "algebraic dynamics",
    "algebraic Z^d-actions",
    "non-mixing sets",
    "sparse multiples of polynomials",
    "shortest polynomial in an ideal",
    "function fields of positive characteristic",
    "Ward's six problems in algebraic dynamics",
    "UnsolvedMath",
    "AMR-042-0001",
    "math.DS",
    "math.NT",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-03",
  "publication_date": "2026-10-03",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-03",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-042-0001/",
  "pdf_url": "https://eulersolve.org/papers/amr-042-0001/paper.pdf?v=b6ec602b9274",
  "doi": "10.5281/zenodo.23128132",
  "zenodo_record_url": "https://zenodo.org/records/23128132",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers Problem A of T. Ward's list 'Six problems in algebraic dynamics' (2006): the exact order of mixing is M = 6 for every prime p. The proof shows that every non-zero multiple of the seven-term polynomial f over an algebraic closure of F_p has at least seven terms; the reduction from mixing to sparse multiples applies published results of Schmidt, Masser and Derksen–Masser. For p = 2 an exhaustive computer search confirms the result for coefficients in F_2. The non-mixing sets with seven points are not determined for odd p. Unrefereed.",
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    "source.zip": {
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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."
}