Algebraic Stories — Non-negative forms

Definitions and mathematical statement

The supplied record uses the symbol $B_{n,3}$ but gives no definition of it, no class of forms over which it is taken, and no explanation of the roles of the indices. It therefore does not determine a particular numerical sequence.

Requested target, conditional on recovering that definition. Determine the value of \[ L=\lim_{n\to\infty}\frac{B_{n,3}}{n^2}. \] For a real-valued sequence indexed by positive integers, determining this limit means finding $L\in\mathbb R$ such that \[ \text{for every }\varepsilon>0\text{ there exists }N\geq1 \text{ such that for every integer }n\geq N, \quad\left|\frac{B_{n,3}}{n^2}-L\right|<\varepsilon. \] The supplied research summary reports that the limit exists and satisfies $5/18\leq L\leq1/2$, with its exact value open. These are attributed claims about the undefined source sequence, not substitutes for its missing definition. The title refers to nonnegative forms; that context alone does not identify $B_{n,3}$.

Short English statement

Find the exact asymptotic constant for the source quantity $B_{n,3}/n^2$. Its definition is absent, so the record supports only a conditional target and the supplied claimed interval $[5/18,1/2]$.

Sources

• Fröberg, Lundqvist, Oneto, and Shapiro, Algebraic Stories from One and from the Other Pockets (2018), source-order Problem 20, https://arxiv.org/abs/1801.01692; source-TeX extraction, accessed 2026-07-29.
• The supplied summary mentions Petrovsky--Oleinik and the neighborhood of Hilbert's seventeenth problem, without an identifiable publication or theorem supporting the limit or bounds.
• Local registry data/unsolvedmath/problems.json, permanent ID $1500020$, source code catalogue entry, Algebraic Stories --- Non-negative forms. The record labels this open and OPEN-TRIAGE; its background marks source-status and rights checks NEEDS_REVIEW.