Breaking the High-Degree Barrier in Lehmer's Problem with a Human-AI Discovery Loop

Forum topic · 小凯 · 2026-10-24

Summary

This arXiv paper (2610.30277) by Yuri Chervonyi, Sergei Gukov, and Thang Luong tackles Lehmer's 1933 problem: whether integer polynomials can have arbitrarily small Mahler measure, which quantifies how far roots stray from the unit circle. Prior systematic constructions covered only even degrees d ≤ 180, and Boyd's analytical one-parameter family frequently factors at higher degrees, skipping nearly 50% of target degrees with no general construction for arbitrary d. The authors introduce a human-AI discovery loop combining conversational prompting with a fast numerical pipeline, an exact symbolic verifier, and large-scale distributed search, where the AI agent prototypes solvers, launches parallel cluster searches, and co-develops hypotheses with human researchers. The result is a five-parameter Generalized Boyd algorithm that eliminates degree-skipping, constructing irreducible polynomials with Mahler measure strictly below Boyd's limit point α₁ ≈ 1.255434 for every target degree. Evaluated on all even degrees up to 2000, it improves six previously published values and holds 99.6% of best known records, with a verified dataset released.

Paper Overview

Field: Machine Learning
Authors: Yuri Chervonyi, Sergei Gukov, Thang Luong
Published: 2026-10-24
arXiv: 2610.30277

Abstract (full translation)

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 $\alpha_1 \approx 1.255434$ 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.

Key Points

Lehmer's problem (1933) asks if the Mahler measure of integer polynomials can be made arbitrarily small.
Previous systematic constructions only covered even degrees d ≤ 180; Boyd's one-parameter family factors at high degrees, skipping ~50% of target degrees.
A human-AI discovery loop combines conversational prompting, a fast numerical pipeline, an exact symbolic verifier, and distributed parallel search.
The resulting five-parameter Generalized Boyd algorithm constructs irreducible polynomials with Mahler measure strictly below α₁ ≈ 1.255434 for every target degree.
Across even degrees up to 2000: 6 published records improved, 99.6% of best known records held; a verified dataset of minimal-Mahler-measure polynomials is released.

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