Algebraic Stories — Non-negative forms

Definitions and mathematical statement

The source's finiteness question is \[ B_{n,m}<\infty\quad\text{and}\quad B'_{n,m}<\infty \qquad\text{for every admissible pair }(n,m)\text{ with }n\text{ even}. \] The two indexed quantities $B_{n,m}$ and $B'_{n,m}$ are not defined anywhere in the supplied record. In particular, the record does not say what objects they bound, what minimum or supremum defines them, whether they concern all nonnegative forms or a restricted class, what the indices count, or which values of $n,m$ are admissible. The title "Non-negative forms" does not determine these definitions. Thus the displayed question faithfully records the intended universal finiteness target, but a self-contained mathematical assertion cannot be recovered without the source definitions. No substitute meaning for either invariant is assigned.

The supplied assessment reports finiteness for $m=2,3$ and for $(n,m)=(4,4)$, attributed to the source label CLR80. It also reports the bound \[ B_{n,m}\leq 2\,\frac{(m-1)^{n+1}-1}{m-2} \] under unspecified hypotheses, attributed to CS13, and says Ko17 shows it is not sharp. The written expression is undefined at $m=2$; no limiting value or replacement formula is inferred. Where valid and finite this bound would establish finiteness of $B_{n,m}$, but the record gives neither its exact scope nor an analogous bound for $B'_{n,m}$. Those omissions must be resolved before reconciling it with the reported remaining general finiteness question.

Short English statement

The question asks whether both source-defined invariants $B_{n,m}$ and $B'_{n,m}$ are finite whenever $n$ is even. Their definitions and index domains are absent, so assigning them a particular nonnegative-form interpretation would invent the target. The supplied assessment labels the problem partly solved, reports the small-parameter cases and a nonsharp bound for $B_{n,m}$, and says the full finiteness statement remains unverified in its literature review. This is a supplied assessment rather than an independent status determination.

Sources

• UnsolvedMath permanent ID 1500018, source code catalogue entry, "Algebraic Stories --- Non-negative forms." Registry curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0; updated 2026-08-06.
• Fröberg, Lundqvist, Oneto, Shapiro, Algebraic Stories from One and from the Other Pockets (2018), source-order Problem 18: https://arxiv.org/abs/1801.01692. Supplied source-TeX extraction accessed 2026-07-29. The metadata marks status evidence and rights review NEEDS_REVIEW and says the source presents an open item.
• The supplied research summary uses bibliography labels CLR80, CS13, and Ko17 without supplying their bibliographic entries. No authors, titles, or locators are inferred for these labels.