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Two-variable polynomials with dynamical Mahler measure zero

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Abstract

We discuss several aspects of the dynamical Mahler measure for multivariate polynomials. We prove a weak dynamical version of the Boyd–Lawton formula, and we characterize the polynomials with integer coefficients having dynamical Mahler measure zero both for the case of one variable (Kronecker’s lemma) and for the case of two variables, under the assumption that the dynamical version of Lehmer’s question is true.

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Notes

  1. Classically the Chebyshev polynomials are normalized so that \({\widetilde{T}}_d(\cos t) = \cos (dt)\). The two normalizations satisfy \({\widetilde{T}}_d(w) = \frac{1}{2} T_d (2w)\).

  2. The fact that the pushforward is only defined up to multiplication by a scalar is not significant, since we only use this construction in the context of order of vanishing.

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Acknowledgements

We are grateful to Patrick Ingram for proposing that we study the dynamical Mahler measure of multivariate polynomials and for many early discussions, and to the anonymous referees for their careful reading of the article and several helpful suggestions. This project was initiated as part of the BIRS workshop “Women in Numbers 5”, held virtually in 2020. We thank the workshop organizers, Alina Bucur, Wei Ho, and Renate Scheidler for their leadership and encouragement that extended for the whole duration of this project. This work has been partially supported by the Natural Sciences and Engineering Research Council of Canada (Discovery Grant 355412-2013 to ML), the Fonds de recherche du Québec - Nature et technologies (Projets de recherche en équipe 256442 and 300951 to ML), the Simons Foundation (grant number 359721 to MM), and the National Science Foundation (grant DMS-1844206 supporting AC and grant DMS-1902772 to LM). This material is based upon work supported by and while the third author served at the National Science Foundation. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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Authors and Affiliations

  1. Department of Mathematics, University of California San Diego, 9500 Gilman Drive # 0112, La Jolla, CA, 92093-0112, USA

    Annie Carter

  2. Département de mathématiques et de statistique, Université de Montréal, CP 6128, succ. Centre-ville, Montreal, QC, H3C 3J7, Canada

    Matilde Lalín

  3. Department of Mathematics, University of Hawaii, 2565 McCarthy Mall, Honolulu, HI, 96822, USA

    Michelle Manes

  4. Mathematical Reviews, 416 Fourth St., Ann Arbor, MI, 48103, USA

    Alison Beth Miller

  5. Department of Mathematics, University of Chicago, 5734 S. University Avenue, Room 108, Chicago, IL, 60637, USA

    Lucia Mocz

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  1. Annie Carter
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  2. Matilde Lalín
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Correspondence to Michelle Manes.

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Carter, A., Lalín, M., Manes, M. et al. Two-variable polynomials with dynamical Mahler measure zero. Res. number theory 8, 25 (2022). https://doi.org/10.1007/s40993-022-00322-z

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