Advances in Theoretical and Mathematical Physics

Volume 27 (2023)

Number 5

Mahler measuring the genetic code of amoebae

Pages: 1405 – 1461

DOI: https://dx.doi.org/10.4310/ATMP.2023.v27.n5.a3

Authors

Siqi Chen (Balliol College, University of Oxford,UK; and London Institute for Mathematical Sciences, London, UK)

Yang-Hui He (London Institute for Mathematical Sciences, London, UK; Dept. of Mathematics, City University of London; Merton College, University of Oxford; and School of Physics, NanKai University, Tianjin, China )

Edward Hirst (London Institute for Mathematical Sciences, London, UK; and Department of Mathematics, City University of London, UK)

Andrew Nestor (London Institute for Mathematical Sciences, London, UK; and Hertford College, University of Oxford, UK)

Ali Zahabi (London Institute for Mathematical Sciences, London, UK; and Institut de Mathématiques de Bourgogne, Université Bourgogne Franche-Comté, France)

Abstract

Amoebae from tropical geometry and the Mahler measure from number theory play important roles in quiver gauge theories and dimer models. Their dependencies on the coefficients of the Newton polynomial closely resemble each other, and they are connected via the Ronkin function. Genetic symbolic regression methods are employed to extract the numerical relationships between the 2d and 3d amoebae components and the Mahler measure. We find that the volume of the bounded complement of a d‑dimensional amoeba is related to the gas phase contribution to the Mahler measure by a degree‑d polynomial, with d = 2 and 3. These methods are then further extended to numerical analyses of the non-reflexive Mahler measure. Furthermore, machine learning methods are used to directly learn the topology of 3d amoebae, with strong performance. Additionally, analytic expressions for boundaries of certain amoebae are given.

Published 15 July 2024