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

In Oberwolfach Report 7/2022 Y. Xu asked whether Gaussian cubature rules of degree 2n − 2, with dim Π³n−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) − δαα′) 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. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-9790358-011 (Xu, Polynomial interpolation and cubature rules, Oberwolfach Rep. 19 (2022), report 7/2022, pp. 415–417).