The Mahler measure of a three-variable family and an application to the Boyd–Lawton formula

Abstract

We prove a formula relating the Mahler measure of an infinite family of three-variable polynomials to a combination of the Riemann zeta function at \(s=3\) and special values of the Bloch–Wigner dilogarithm by evaluating a regulator. The evaluation requires two different applications of Jensen’s formula and analyzing the integral in two different planes, as opposed to the more common strategy of using only one plane. The degrees of the monomials involving one of the variables are allowed to vary freely, leading to an interesting application of the Boyd–Lawton formula.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2

References

  1. 1.

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

    MathSciNet  Article  Google Scholar 

  2. 2.

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

    MathSciNet  Article  Google Scholar 

  3. 3.

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

    MathSciNet  Article  Google Scholar 

  4. 4.

    Boyd, D.W.: Conjectural explicit formulas for the Mahler measure of some three variable polynomials. Unpublished Notes (2006)

  5. 5.

    Boyd, D.W., Mossinghoff, M.J.: Small limit points of Mahler’s measure. Exp. Math. 14(4), 403–414 (2005)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Boyd, D.W., Rodriguez-Villegas, F.: Mahler’s measure and the dilogarithm. I. Can. J. Math. 54(3), 468–492 (2002)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Boyd, D.W., Rodriguez-Villegas, F., Dunfield, N.M.: Mahler’s measure and the dilogarithm II. arXiv:math/0308041 (2003)

  8. 8.

    D’Andrea, C., Lalín, M.N.: On the Mahler measure of resultants in small dimensions. J. Pure Appl. Algebra 209(2), 393–410 (2007)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Darboux, G.: Mémoire sur les fonctions discontinues. Ann. Sci. École Norm. Sup. (2) 4, 57–112 (1875)

    MathSciNet  Article  Google Scholar 

  10. 10.

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

    MathSciNet  Article  Google Scholar 

  11. 11.

    Goncharov, A.B.: Regulators. In: Handbook of $K$-Theory, vols. 1, 2. Springer, Berlin, pp. 295–349 (2005)

  12. 12.

    Guillemin, V., Pollack, A.: Differential Topology. AMS Chelsea Publishing, Providence. Reprint of the 1974 original (2010)

  13. 13.

    Lalín, M.N.: An algebraic integration for Mahler measure. Duke Math. J. 138(3), 391–422 (2007)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Lalín, M.N.: Mahler measures and computations with regulators. J. Number Theory 128(5), 1231–1271 (2008)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Lawton, W.M.: A problem of Boyd concerning geometric means of polynomials. J. Number Theory 16(3), 356–362 (1983)

    MathSciNet  Article  Google Scholar 

  16. 16.

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

    MathSciNet  Article  Google Scholar 

  17. 17.

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

    MathSciNet  Article  Google Scholar 

  18. 18.

    Maillot, V.: Mahler measure in Arakelov geometry. In: Workshop Lecture at “ The Many Aspects of Mahler’s Measure”. Banff International Research Station, Banff, Canada (2003)

  19. 19.

    Rodriguez-Villegas, F.: Modular Mahler Measures. I. Topics in Number Theory (University Park, PA, 1997). Math. Appl., vol. 467, pp. 17–48. Kluwer, Dordrecht (1999)

  20. 20.

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

    MathSciNet  Article  Google Scholar 

  21. 21.

    Zagier, D.: Polylogarithms, Dedekind, zeta functions and the algebraic $K$-theory of fields. In: Arithmetic Algebraic Geometry (Texel, 1989). Progr. Math., vol. 89, pp. 391–430. Birkhäuser Boston, Boston (1991)

Download references

Acknowledgements

The authors are grateful to David Boyd and Mathew Rogers for some early discussions in the topic of these formulas and would like to thank the referee for helpful suggestions and corrections.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jarry Gu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported by the Natural Sciences and Engineering Research Council of Canada [Discovery Grant 355412-2013], the Fonds de recherche du Québec - Nature et technologies [Projet de recherche en équipe 256442], and the Institut des sciences mathématiques.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Gu, J., Lalín, M. The Mahler measure of a three-variable family and an application to the Boyd–Lawton formula. Res. number theory 7, 13 (2021). https://doi.org/10.1007/s40993-021-00237-1

Download citation

Keywords

  • Mahler measure
  • Boyd–Lawton formula
  • Regulator

Mathematics Subject Classification

  • Primary 11R06
  • Secondary 11M06
  • 11R42