Human and AI team up to find new polynomials with smaller Mahler measure
Breaking the high-degree barrier in Lehmer's problem with a human-AI discovery loop
Artificial Intelligence
Summary
Lehmer’s problem asks if certain math objects called integer polynomials can have measures (called Mahler measure) that get arbitrarily close to one. The new work uses a loop where humans and AI work together, combining fast numerical methods, exact checks, and large-scale searches to find new constructions of these polynomials. Their approach works for very high degrees, beyond what was done before, and finds many records. This collaboration produces a verified dataset of polynomials with the smallest known Mahler measures for many cases.
What this means in practice
- •For mathematical software developers: Improve computer algebra systems by integrating AI-driven polynomial discovery methods for exploring integer polynomial properties.
- •For cryptography engineers: Use newly constructed irreducible polynomials with controlled Mahler measure in algorithms that depend on algebraic number theory structures.
Authors
Yuri Chervonyi, Sergei Gukov, Thang Luong
Abstract
While artificial intelligence has made rapid progress in mathematical reasoning, many open problems depend on structure that is revealed only through large-scale computational exploration and analytical synthesis. One such example is Lehmer's problem from 1933. It asks whether the Mahler measure of an integer polynomial, which measures how far its roots stray from the unit circle, can be made arbitrarily small. Previously, constructing irreducible polynomials with integer coefficients and small Mahler measure has been carried out systematically only for even degrees $d \le 180$. At higher degrees, Boyd's analytical one-parameter family frequently factors, skipping nearly 50% of target degrees and leaving no general construction for arbitrary $d$. Here, we address the high-degree regime through a human-AI discovery loop, in which we extend the conversational prompting with a fast numerical pipeline, an exact symbolic verifier, and large-scale distributed search. The AI agent prototypes numerical and symbolic solvers locally, launches parallel searches across distributed clusters and works with the human researcher to extract and verify mathematical hypotheses. Through this interactive loop, we derive a five-parameter construction, which we call the Generalized Boyd algorithm. It eliminates degree-skipping and constructs irreducible polynomials with Mahler measure strictly below Boyd's limit point $α_1 \approx 1.255434$ for every target degree. Evaluated across all even degrees up to 2000, our approach improves upon six previously published values and holds 99.6% of all best known records. With this work we release the verified dataset of polynomials with minimal Mahler measure across all studied even degrees.