Low-degree polynomial parametrizations of uv(u+v)=g³+g²+3g−1
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Can rational polynomial families u,v,g satisfy u(t)v(t)(u(t)+v(t))=g(t)³+g(t)²+3g(t)−1 with nonconstant g? The recorded exclusion concerns degree(g)≤4, not all polynomial degrees or all rational solutions.
Why it matters and what counts as progress
Why it matters: A bounded-degree exclusion closes one route to parametrizing the cubic equation without settling unrestricted polynomial or rational solutions.
What counts as progress: Exclude additional polynomial degrees or construct a valid family beyond the known exclusion range.
How the problem developed
- 26 Aug 2026 · Formalized nonexistence [1]
Claude Code agents, under Brandon M. Yates’s direction Reported a Lean-checked exclusion of rational polynomial triples with nonconstant g of degree 1 through 4 satisfying the displayed equation; Claude agents developed the argument and independently re-verified the formalization.
AI contribution
Claude-led proof excludes low-degree cubic polynomial families
For nonconstant g∈Q[t] of degree at most four, no u,v∈Q[t] satisfy u(t)v(t)(u(t)+v(t))=g(t)³+g(t)²+3g(t)−1. This excludes only those low-degree polynomial families.
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Publication dates and editorial changes.
Claude-led proof excludes low-degree cubic polynomial families