[论文] Breaking the high-degree barrier in Lehmer's problem with a human-AI d...

论文概要

研究领域: ML 作者: Yuri Chervonyi, Sergei Gukov, Thang Luong 发布时间: 2026-10-24 arXiv: 2610.30277

中文摘要

Lehmer 1933 年问题问：整数多项式的 Mahler 测度——刻画其根偏离单位圆的程度——能否任意小。此前系统构造仅覆盖偶数度 d≤180；Boyd 的解析单参数族在高次数频繁因式分解，跳过近 50% 的目标次数，对任意 d 没有通用构造。本文通过 human-AI 发现循环攻克高度数区间：把对话式提示与快速数值流水线、精确符号验证器和大规模分布式搜索结合——AI 智能体在本地原型化数值与符号求解器，跨分布式集群发起并行搜索，并与人类研究者一起提炼、验证数学假设。通过这一交互循环导出五参数构造，称为广义 Boyd 算法：消除次数跳过，对每个目标度数构造 Mahler 测度严格低于 Boyd 极限点 α₁≈1.255434 的不可约多项式。在 2000 以内全部偶数度上评估，改进了 6 个已发表数值，保持 99.6% 的历史最优纪录，并发布全部研究偶数度的经验证最小 Mahler 测度多项式数据集。

原文摘要

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.

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*自动采集于 2026-10-24*

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