Degree-4 polynomials evaluated with two multiplications as Lean-verified old proof finally published

thomasahle published a result he proved by hand years ago but never dared to submit due to its hundred-plus-page length: a degree-4 polynomial P(x)=x⁴+a₃x³+a₂x²+a₁x+a₀ can be evaluated with only two multiplications. He stresses the math involved no AI at all—AI only verified it, via formalization in the Lean proof assistant, which gave him the confidence to publish. srchvrs relayed the result in a reply.

Confirmed

• Horner's method evaluates a degree-n polynomial with n multiplications; with rational preprocessing, a monic polynomial needs only ⌊n/2⌋+1 multiplications (one more for general polynomials).
• The trick writes the polynomial in nested form, e.g. computing an intermediate y = (x + b₀)x + b₁ and combining from there.
• A degree-4 polynomial thus reduces to two multiplications, roughly doubling evaluation speed.
• The proof was done by hand years ago; publication was triggered by successful Lean formal verification.

Why it matters

• The case demonstrates the practical value of formal verification tools like Lean for mathematical publishing: they remove trust concerns over very long proofs and let shelved results see the light.
• The author explicitly distinguishes "AI doing math" from "AI verifying math," offering a clear example of AI's role in mathematical research.

2026-09-11 ~ 2026-09-11 · 5 related posts

• [source] AI-Verified Lean Proof Shows Quartic Polynomials Need Only Two Multiplications — thomasahle · 2026-09-11
• [source] Polynomials evaluated in just two multiplications, verified in Lean with AI's help after years unpublished — thomasahle · 2026-09-11
• Polynomial evaluation in just two multiplications: 100-page proof finally verified in Lean by AI — srchvrs · 2026-09-11

2 near-duplicate retellings: thomasahle · thomasahle