The zero truncated product on P_K is hyperbolic exactly when the mean is not a Gauss node: the?' cells of the associative-truncated-product table

Abstract

Meghaichi and Xing (arXiv:2606.12632) discretise nonlinear conservation laws with uncertainty by replacing the multiplication on the space P_K of polynomials of degree at most K in the random variable by an associative truncated product (ATP), and they tabulate five such products against four properties; three cells of their table are printed?'. We observe that every ATP on P_K is the quotient product past q=pq mod ω for a unique monic ω of degree K+1, with ω=φ₁φ_K-φ₁astφ_K up to a scalar, so that hyperbolicity, symmetry and positive definiteness become statements about the roots of one polynomial. For the zero product, ω=π₁π_K, the two?' cells are settled exactly: it is hyperbolic if and only if π_K(μ) ≠ 0, that is, iff the mean μ of the weight is not a node of its K-point Gauss rule — a repeated root of ω always produces a Jordan chain — and it is positive definite whenever μ and the Gauss nodes lie in Ξ, whether or not it is hyperbolic, because the eigenvalues of every multiplication operator are the values of the multiplier at the roots of ω with no diagonalisability assumption. Consequently K=1 is never hyperbolic and K=2 always (π₂(μ)=-Var w), for a symmetric weight hyperbolicity is equivalent to K even, and the two cells are independent: the Legendre weight at K=3 is defective but positive definite, while the uniform weight on [-2,-1] ∪ [1,2] at K=2 and an explicit four-point weight at K=3 (an odd K that is hyperbolic) are hyperbolic but not positive definite. The third?' cell follows from the same dictionary and a lemma of the source: the collocation product at K+1 distinct nodes is symmetric iff the nodes are the zeros of π_(K+1)+t π_K for some real t. The two criteria, the cells and the controls are verified in Lean 4 against Mathlib, without compiled evaluation.

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