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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.

References

  1. Benedetto, R., Ingram, P., Jones, R., Manes, M., Silverman, J.H., Tucker, T.J.: Current trends and open problems in arithmetic dynamics. Bull. Am. Math. Soc. (N.S.) 56(4), 611–685 (2019)

    Article  MathSciNet  Google Scholar 

  2. Boyce, W.M.: On polynomials which commute with a given polynomial. Proc. Am. Math. Soc. 33, 229–234 (1972)

    Article  MathSciNet  Google Scholar 

  3. Boyd, D.W.: Kronecker’s theorem and Lehmer’s problem for polynomials in several variables. J. Number Theory 13(1), 116–121 (1981)

    Article  MathSciNet  Google Scholar 

  4. Boyd, D.W.: Speculations concerning the range of Mahler’s measure. Can. Math. Bull. 24(4), 453–469 (1981)

    Article  MathSciNet  Google Scholar 

  5. Boyd, D.W.: Mahler’s measure and special values of \(L\)-functions. Exper. Math. 7(1), 37–82 (1998)

    Article  MathSciNet  Google Scholar 

  6. Breusch, R.: On the distribution of the roots of a polynomial with integral coefficients. Proc. Am. Math. Soc. 2, 939–941 (1951)

    Article  MathSciNet  Google Scholar 

  7. Brolin, H.: Invariant sets under iteration of rational functions. Ark. Mat. 6(1965), 103–144 (1965)

    Article  MathSciNet  Google Scholar 

  8. Carter, A., Lalín, M.,Manes, M., Miller, A.B.: Dynamical Mahler Measure: A survey and some recent results (in preparation)

  9. Chambert-Loir, A., Thuillier, A.: Mesures de Mahler et équidistribution logarithmique. Ann. Inst. Fourier (Grenoble) 59(3), 977–1014 (2009)

    Article  MathSciNet  Google Scholar 

  10. Gregory, S., Silverman, C., Joseph, H.: Canonical heights on varieties with morphisms. Compos. Math. 89(2), 163–205 (1993)

    MathSciNet  MATH  Google Scholar 

  11. Deninger, C.: Deligne periods of mixed motives, \(K\)-theory and the entropy of certain \({\mathbf{Z}}^n\)-actions. J. Am. Math. Soc. 10(2), 259–281 (1997)

    Article  Google Scholar 

  12. Dimitrov, V.: A proof of the Schinzel-Zassenhaus conjecture on polynomials. arXiv:1912.12545

  13. Dobrowolski, E.: On a question of Lehmer and the number of irreducible factors of a polynomial. Acta Arith. 34(4), 391–401 (1979)

    Article  MathSciNet  Google Scholar 

  14. Everest, G., Ward, T.: Heights of Polynomials and Entropy in Algebraic Dynamics, p. 1700272. Springer, London (1999)

    Book  Google Scholar 

  15. Favre, C., Rivera-Letelier, J.: Équidistribution quantitative des points de petite hauteur sur la droite projective. Math. Ann. 335(2), 311–361 (2006)

    Article  MathSciNet  Google Scholar 

  16. Freire, A., Lopes, A., Mañé, R.: An invariant measure for rational maps. Bol. Soc. Brasil. Mat. 14(1), 45–62 (1983)

    Article  MathSciNet  Google Scholar 

  17. Ghioca, D.,Tucker, T.J., Zhang, S.: Towards a dynamical Manin-Mumford conjecture. Int. Math. Res. Not. IMRN (22) 5109–5122 (2011)

  18. Ghioca, D., Nguyen, K.D., Ye, H.: The dynamical Manin-Mumford conjecture and the dynamical Bogomolov conjecture for split rational maps. J. Eur. Math. Soc. (JEMS) 21(5), 1571–1594 (2019)

    Article  MathSciNet  Google Scholar 

  19. Ingram, P.: Variation of the canonical height for a family of polynomials. J. Reine Angew. Math. 685, 73–97 (2013)

    MathSciNet  MATH  Google Scholar 

  20. Julia, G.: Mémoire sur la permutabilité des fractions rationnelles. Ann. Sci. École Norm. Sup. 39(3), 131–215 (1922)

    Article  MathSciNet  Google Scholar 

  21. Kronecker, L.: Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten. J. Reine Angew. Math. 53, 173–175 (1857)

    MathSciNet  Google Scholar 

  22. Lawton, W.M.: J. Number Theory 16(3), 356–362 (1983)

    Article  MathSciNet  Google Scholar 

  23. Lehmer, D.H.: Factorization of certain cyclotomic functions. Ann. Math. (2) 34(3), 461–479 (1933)

    Article  MathSciNet  Google Scholar 

  24. Lyubich, M.J.: Entropy properties of rational endomorphisms of the Riemann sphere. Ergodic Theory Dyn. Syst. 3(3), 351–385 (1983)

    Article  MathSciNet  Google Scholar 

  25. Mahler, K.: On some inequalities for polynomials in several variables. J. Lond. Math. Soc. 37, 341–344 (1962)

    Article  MathSciNet  Google Scholar 

  26. Pineiro, J., Szpiro, L., Tucker, T.J.: Mahler measure for dynamical systems on \({\mathbb{P}}^1\) and intersection theory on a singular arithmetic surface, Geometric methods in algebra and number theory. Progr. Math. 235, 219–250 (2005)

    Article  Google Scholar 

  27. Ransford, T.: Potential Theory in the Complex Plane. London Mathematical Society Student Texts, vol. 28, p. 1334766. Cambridge University Press, Cambridge (1995)

    Book  Google Scholar 

  28. Ritt, J.F.: Permutable rational functions. Trans. Am. Math. Soc. 25(3), 399–448 (1923)

    Article  MathSciNet  Google Scholar 

  29. Rogers, M., Zudilin, W.: On the Mahler measure of \(1+X+1/X+Y+1/Y\). Int. Math. Res. Not. IMRN (9), 2305–2326 (2014)

  30. Silverman, H.J.: The Arithmetic of Dynamical Systems. Graduate Texts in Mathematics, vol. 241, p. 2316407. Springer, New York (2007)

    Book  Google Scholar 

  31. Smyth, C.J.: On the product of the conjugates outside the unit circle of an algebraic integer. Bull. Lond. Math. Soc. 3, 169–175 (1971)

    Article  MathSciNet  Google Scholar 

  32. Smyth, C.J.: On measures of polynomials in several variables. Bull. Austr. Math. Soc. 23(1), 49–63 (1981)

    Article  MathSciNet  Google Scholar 

  33. Szpiro, L., Tucker, T.J.: Equidistribution and Generalized Mahler Measures, Number Theory, Analysis and Geometry, pp. 609–638. Springer, New York (2012)

    Book  Google Scholar 

  34. Zhang, S.: Small points and Adelic metrics. J. Algebr. Geom. 4(2), 281–300 (1995)

    MathSciNet  MATH  Google Scholar 

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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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