The arithmetic of Boyd’s Mahler measure conjectures: from data analysis to machine learning

Abstract: The Mahler measure is an invariant of multivariable rational functions defined by averaging $\log |P|$ over the unit torus. Boyd conjectured that, for \[ x+y+\frac{1}{x}+\frac{1}{y}+k, \] its Mahler measure is a rational multiple $r_k L'(E_k,0)$, where $E_k$ is an associated elliptic curve. We study the arithmetic of the factors $r_k$ using a dataset of the first 250,000 values of $k$, combining large-scale statistical analysis with transformer-based experiments. We observe several striking patterns in their size and $p$-adic behavior, and find that the neural networks recover substantial arithmetic structure from the data, and perhaps a bit more. This is joint work with Alberto Alfarano, Pablo Bianucci, and Berend Ringeling.

computation and languagemachine learningcombinatoricsnumber theory

Audience: researchers in the topic

AI, Combinatorics and Number Theory Seminar

Series comments: This seminar is devoted to the responsible and constructive use of AI in number theory and combinatorics. It is a space to think carefully about what these tools can and cannot do, and where the limits actually lie. Enthusiasm and skepticism are both welcome here, and we ask only that they be voiced with generosity. Questions from students and from people new to these areas are especially encouraged.

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