All-degree positive cubature at the dimension-matched scale

Construct positive degree-2n cubature rules on the truncated conic surface with node count N=D_n+n+O(1), where D_n=(n+1)^2=dim\Pi_n(V), for all degrees n, or otherwise determine whether the dimension-matched scale suggested by the Prony-type formulation is attainable.

Background

The quadratic disk-to-cone quotient converts positive degree-2n cone cubature into centrally symmetric positive degree-(4n+1) disk cubature. In an origin-containing orbit family, a parameter-count argument shows that if the number of radial orbit parameters satisfies I=n-1+O(1) and the number of scalar unknowns is D_n+O(1), then the resulting cone rule would have N=D_n+n+O(1) nodes.

The proposition is explicitly conditional and does not establish solvability of the nonlinear moment system subject to disk containment and positive weights. Moreover, Möller’s lower bound is not always attained: the paper notes that a 17-point degree-nine disk rule does not exist, although this does not rule out every possible nine-node cone rule. The unresolved issue is therefore existence of positive rules at the proposed scale for all degrees, rather than merely the parameter count.

References

Two questions remain open. The first is the weighted sharp law suggested by eq:weighted-conjecture. The second is the existence of all-degree positive rules at the dimension-matched scale D_n+n+O(1) suggested by \Cref{prop:Prony}.

eq:weighted-conjecture:

Λn,γV,∏≍{n1/2,−1<γ<−1/2,nγ+1,γ≥−1/2.\Lambda_{n,\gamma}^{V,\prod}\asymp \begin{cases} n^{1/2},&-1<\gamma<-1/2,\\ n^{\gamma+1},&\gamma\ge-1/2. \end{cases}

— Positive Cubature Compression and Optimal Hyperinterpolation Stability on a Conic Surface  (2609.36430 - An et al., 29 Sep 2026) in Section 8, Section “Conclusions”; Proposition (Prony) and the final paragraph