{
  "schema_version": 1,
  "problem_number": "OWR-9790358-011",
  "title": "Gaussian Cubature of Even Degree for the Product Chebyshev Weight in Three Variables: An Explicit Ten-Node Rule and No Twenty-Node Rule with a Sign Symmetry",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "In Oberwolfach Report 7/2022 Y. Xu asked whether Gaussian cubature rules of degree 2n − 2, with dim Π³_{n−1} = (n+2 choose 3) real nodes, exist for the product Chebyshev weight of the second kind on the cube [−1,1]³; existence is known only for n = 2 and n = 3. (The source prints the formula of the weight of the first kind under the name of the second kind; we follow the name.) We give partial results. From Xu's characterisation of Gaussian rules of even degree we derive that, for this weight in d variables, such a rule exists if and only if an explicit quadratic system has a real solution: for a function h on the multi-indices of length 2n − 1, the matrix (Σ_β h(α+β) h(α′+β) − δ_{αα′}) must depend only on α + α′; the rules of Morrow and Patterson in two variables correspond to h = δ_{(n,n−1)}. For n = 3 we give an explicit ten-node rule with nodes and weights in Q(√2, √3). For n = 4 we prove that no Gaussian rule of degree 6 has a node set invariant under a coordinate reflection, a half-turn about a coordinate axis, or x ↦ −x, hence under any non-trivial group of coordinate sign changes. The cases of one reflection and of a half-turn are computer-assisted, by exact Nullstellensatz certificates with rational coefficients, checked by independently written programs; two reflections and the central symmetry have short direct proofs. The case n = 4 remains open for node sets whose symmetry group contains no non-trivial sign change (for example a group generated by (x, y, z) ↦ (−y, −x, z) or by a rotation of order three, or the trivial group), and so do all n ≥ 5; numerical searches for n = 4 and n = 5 found no rule, which is evidence only. The question and the theory of Gaussian cubature of even degree are due to Xu, building on Morrow, Patterson and Schmid; new, to our knowledge, are the exclusion of coordinate reflections and half-turns for n = 4 and the explicit rule for n = 3. This is an unrefereed note.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.NA",
    "math.CA",
    "math.AC"
  ],
  "keywords": [
    "Gaussian cubature",
    "cubature rules of even degree",
    "product Chebyshev weight",
    "Chebyshev polynomials of the second kind",
    "quasi-orthogonal polynomials",
    "multi-Hankel matrices",
    "Nullstellensatz certificates",
    "computer-assisted proof",
    "partial answer",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-9790358-011",
    "math.NA",
    "math.CA",
    "math.AC",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-15",
  "publication_date": "2026-10-15",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-15",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-9790358-011/",
  "pdf_url": "https://eulersolve.org/papers/owr-9790358-011/paper.pdf?v=108289985f0e",
  "doi": "10.5281/zenodo.23399617",
  "zenodo_record_url": "https://zenodo.org/records/23399617",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "On Xu's question (Oberwolfach Report 7/2022) on Gaussian cubature of degree 2n-2 for the product Chebyshev weight on the cube: a criterion by a quadratic multi-Hankel system, an explicit ten-node rule for n = 3, and for n = 4 no rule whose node set is invariant under a non-trivial group of coordinate sign changes; the question for n at least 4 remains open.",
  "files": {
    "paper.pdf": {
      "sha256": "108289985f0e8896551c66277b3cebcf1ced0a4d4637b38869ed59dde324c767"
    },
    "source.zip": {
      "sha256": "14890f787f68723982f7d7566c40f62568c714ee371c6c4cb3481ad4ee910f3c"
    },
    "verification_report.md": {
      "sha256": "4d0962f79c97b73cc5ae259d319d1c3d808aa2e518b94f2314e6f65957c9453d"
    }
  },
  "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."
}