机器学习博伊德（Boyd）马勒测度猜想的算术性质
Machine learning the arithmetic of Boyd's Mahler measure conjectures
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中文总结 AI 辅助
该研究利用机器学习与统计分析，探究博伊德马勒测度猜想中有理数$r_k$的算术性质，重现关键观测并发现素数2、3下的特殊算术结构。
中文摘要 AI 辅助
博伊德猜想，对于整数$k$，多项式$P_k(x,y)=x+y+\frac{1}{x}+\frac{1}{y}+k$的马勒测度可表示为$r_kL'(E_k,0)$，其中$E_k$是与$P_k$零点轨迹关联的椭圆曲线，$r_k$为有理数。我们利用包含前250000个$k$值的数据集，结合Claude辅助的大规模统计分析与Axolver开展的基于Transformer的实验，研究$r_k$的各类算术性质。我们重现了博伊德的观测结果：除少数例外，$r_k$是某个整数的倒数，该整数的大小由椭圆曲线的导子决定。此外，其$p$进赋值随素数呈现显著不同的行为：对于$p\geq5$，$v_p(r_k)=-m$（$m\geq1$）的概率似乎为$p^{-m}$；而对于素数2和3，我们发现涉及$k$的同余条件与$E_k$的坏约化素数的额外算术结构。尽管神经网络无法精确预测$r_k$，但它们能获取其幅值与赋值的重要信息，特别是素数2的实验表明，存在超出我们统计分析所得显式预测器的算术结构。
英文摘要
Boyd conjectured that the Mahler measure of $P_k(x,y)=x+y+\frac{1}{x}+\frac{1}{y}+k$ for $k$ an integer, is given by $r_kL'(E_k,0)$, where $E_k$ is the elliptic curve associated to the zero locus of $P_k$ and $r_k$ is a rational number. We study various arithmetic properties of $r_k$ using a dataset containing the first $250{,}000$ values of $k$, combining large-scale statistical analysis assisted by Claude with transformer-based experiments carried out using Axolver. We recover Boyd's observation that, apart from a few exceptions, $r_k$ is the reciprocal of an integer. The size of this integer is governed by the conductor of the elliptic curve. Moreover, its $p$-adic valuations display markedly different behavior according to the prime. For $p\geq 5$, the probability of $v_p(r_k)=-m$ for $m\geq 1$ appears to be $p^{-m}$. For the primes $2$ and $3$, however, we find additional arithmetic structure involving congruence conditions on $k$ and the primes of bad reduction of $E_k$. Although the neural networks do not predict $r_k$ exactly, they recover significant information about its magnitude and valuations. In particular, the experiments at the prime $2$ suggest arithmetic structure beyond the explicit predictor obtained from our statistical analysis.
