A search for unusually small polynomial measures has produced a more flexible mathematical recipe. Yuri Chervonyi, Sergei Gukov and Thang Luong used an AI agent, distributed computation and exact symbolic checks to develop the Generalized Boyd algorithm, extending a construction connected to Lehmer’s long-standing problem.[1]
Their 21 October preprint describes a loop in which the agent helped build numerical and symbolic solvers, run parallel searches and formulate hypotheses with human researchers. The useful output was both a collection of candidate polynomials and a compact construction explaining patterns in that collection.[1]
Why the degree matters
The degree is the highest power in a polynomial. Earlier systematic searches for small-measure, irreducible integer polynomials reached even degree 180. At higher degrees, Boyd’s one-parameter family often factors into simpler polynomials, so it skips nearly half the desired target degrees. The new five-parameter construction removes that obstacle; the team evaluated its results across even degrees up to 2,000.[1]
What the reported benchmark covers
| Measure | Reported result |
|---|---|
| Degree range evaluated | Even degrees through 2,000 |
| Previously published values improved | Six |
| Share of benchmark best-known records held | 99.6% |
| Construction threshold | Below Boyd’s limit point, approximately 1.255434 |
Lehmer’s original question, posed in 1933, asks whether an integer polynomial’s Mahler measure can approach one arbitrarily closely from above. The measure combines the leading coefficient with contributions from roots outside the unit circle. His degree-ten example has measure about 1.17628. Constructing high-degree examples below 1.255434 expands the available families without beating that famous value or resolving the lower-bound question.[1]
The two kinds of verification
The checking machinery addresses two distinct questions. Symbolic routines test whether a polynomial factors exactly. Numerical root calculations estimate its Mahler measure. Keeping those checks separate matters: exact irreducibility does not automatically make every reported decimal an error-bounded theorem. The released dataset lets other researchers inspect the candidates and repeat the calculations.[1]
The construction also makes the search easier to understand. A large computational sweep can reveal useful examples while leaving their common structure obscure. Here the researchers used the returned patterns to enlarge a known family, giving future searches a more systematic starting point.[1]
The new recipe changes which high-degree examples researchers can readily construct. Lehmer’s question remains open, but work on it now has a denser set of verified candidates and a clearer way to explore beyond the range where earlier searches were systematic.[1]
Sources & further reading
- Breaking the high-degree barrier in Lehmer's problem with a human-AI discovery loop arXiv · 21 Oct 2026 · Accessed 26 Oct 2026