## Mahler's measure for polynomials in several variables

##### Doc. Math. Extra Vol. Mahler Selecta, 45-56 (2019)
DOI: 10.25537/dm.2019.SB-45-56

### Summary

If $P(x_1, \ldots, x_k)$ is a polynomial with complex coefficients, the Mahler measure of $P$, $M(P)$, is defined to be the geometric mean of $|P|$ over the $k$-torus, $\mathbb{T}^k$. We briefly describe Mahler's motivation for defining this function and his applications of it to polynomial inequalities. We then describe how this function occurs naturally in the study of Lehmer's problem concerning the set of all measures of one-variable polynomials with integer coefficients. We describe work of Deninger which shows how Mahler measure arises in the study of the far-reaching Beĭlinson conjectures and leads to surprising conjectural explicit formulas for some measures of multivariable polynomials. Finally we describe some of the recent work of many authors proving some of these formulas by a variety of different methods.

### Mathematics Subject Classification

11-03, 11R06, 11R09

### Keywords/Phrases

Lehmer's problem, Beĭlinson conjectures

### References

• [M148]. K. Mahler, On some inequalities for polynomials in several variables, J. London Math. Soc. 37 (1962), 341-344.
• [M153]. K. Mahler, A remark on a paper of mine on polynomials, Illinois J. Math. 8 (1964), 1-4.
• 1. Marie-José Bertin, Une mesure de Mahler explicite, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), no. 1, 1-3.
• 2. Marie-José Bertin, Mesure de Mahler d'hypersurfaces $K3$, J. Number Theory 128 (2008), no. 11, 2890-2913.
• 3. Marie-José Bertin, Amy Feaver, Jenny Fuselier, Matilde Lalín, and Michelle Manes, Mahler measure of some singular $K3$-surfaces, David, Chantal (ed.) et al., Women in numbers 2: research directions in number theory, Contemp. Math., vol. 606, Amer. Math. Soc., Providence, RI, 2013, pp. 149-169.
• 4. Marie-José Bertin and Matilde Lalín, Mahler measure of multivariable polynomials, David, Chantal (ed.) et al., Women in numbers 2: research directions in number theory, Contemp. Math., vol. 606, Amer. Math. Soc., Providence, RI, 2013, pp. 125-147.
• 5. David W. Boyd, Small Salem numbers, Duke Math. J. 44 (1977), no. 2, 315-328.
• 6. David W. Boyd, Pisot and Salem numbers in intervals of the real line, Math. Comp. 32 (1978), no. 144, 1244-1260.
• 7. David W. Boyd, Reciprocal polynomials having small measure, Math. Comp. 35 (1980), no. 152, 1361-1377.
• 8. David W. Boyd, Speculations concerning the range of Mahler's measure, Canad. Math. Bull. 24 (1981), no. 4, 453-469.
• 9. David W. Boyd, The asymptotic behaviour of the binomial circulant determinant, J. Math. Anal. Appl. 86 (1982), no. 1, 30-38.
• 10. David W. Boyd, Reciprocal polynomials having small measure. II, Math. Comp. 53 (1989), no. 187, 355-357, S1-S5.
• 11. David W. Boyd, Two sharp inequalities for the norm of a factor of a polynomial, Mathematika 39 (1992), no. 2, 341-349.
• 12. David W. Boyd, Mahler's measure and special values of $L$-functions, Experiment. Math. 7 (1998), no. 1, 37-82.
• 13. David W. Boyd, Mahler's measure, hyperbolic geometry and the dilogarithm, CMS Notes 34 (2002), 3-4 and 26-28, Note that the rational numbers that appear in formulas near the end of this paper were incorrectly typeset, so for example in equation (15) of that paper, $72$ should be read as $7/2$ and $34$ should be read as $3/4$.
• 14. David W. Boyd, Mahler's measure and l-functions of elliptic curves at $s = 3$, Slides of a lecture at the Simon Fraser University Number Theory Seminar available at http://www.math.ubc.ca/$\sim$ boyd/sfu06.ed.pdf, 2006.
• 15. David W. Boyd, Mahler's measure and special values of l-functions, Slides of a lecture at the Pacific Northwest Number Theory Seminar in Eugene, Oregon, available at http://www.math.ubc.ca/$\sim$ boyd/pnwnt2015.ed.pdf, 2015.
• 16. David W. Boyd, Christopher Deninger, Douglas Lind, and Fernando Rodriguez Villegas, The many aspects of Mahler's measure, Banff International Research Station Report available at http://www.birs.ca/workshops/2003/03w5035/report03w5035.pdf, 2003.
• 17. David W. Boyd, Nathan M. Dunfield, and Fernando Rodriguez Villegas, Mahler's measure and the dilogarithm. II, available at https://arxiv.org/pdf/math/0308041.pdf, 2003.
• 18. David W. Boyd and Michael J. Mossinghoff, Small limit points of Mahler's measure, Experiment. Math. 14 (2005), no. 4, 403-414.
• 19. David W. Boyd and Fernando Rodriguez Villegas, Mahler's measure and the dilogarithm. I, Canad. J. Math. 54 (2002), no. 3, 468-492.
• 20. Robert Breusch, On the distribution of the roots of a polynomial with integral coefficients, Proc. Amer. Math. Soc. 2 (1951), 939-941.
• 21. François Brunault, Regulators of Siegel units and applications, J. Number Theory 163 (2016), 542-569.
• 22. John D. Condon, Asymptotic expansion of the difference of two Mahler measures, J. Number Theory 132 (2012), no. 9, 1962-1983.
• 23. Christopher Deninger, Deligne periods of mixed motives, $K$-theory and the entropy of certain ${\mathbf Z}^n$-actions, J. Amer. Math. Soc. 10 (1997), no. 2, 259-281.
• 24. J. Dufresnoy and Ch. Pisot, Etude de certaines fonctions méromorphes bornées sur le cercle unité. Application à un ensemble fermé d'entiers algébriques, Ann. Sci. Ecole Norm. Sup. (3) 72 (1955), 69-92.
• 25. Alain Durand, On Mahler's measure of a polynomial, Proc. Amer. Math. Soc. 83 (1981), no. 1, 75-76.
• 26. J. S. Frame, Factors of the binomial circulant determinant, Fibonacci Quart. 18 (1980), no. 1, 9-23.
• 27. A. O. Gel'fond, Transcendental and algebraic numbers, Translated from the first Russian edition by Leo F. Boron, Dover Publications, Inc., New York, 1960.
• 28. Matilde Lalín, Some examples of Mahler measures as multiple polylogarithms, J. Number Theory 103 (2003), no. 1, 85-108.
• 29. Matilde Lalín, Mahler measure and elliptic curve $L$-functions at $s=3$, J. Reine Angew. Math. 709 (2015), 201-218.
• 30. Matilde Lalín and Mathew D. Rogers, Functional equations for Mahler measures of genus-one curves, Algebra Number Theory 1 (2007), no. 1, 87-117.
• 31. Matilde Lalín, Detchat Samart, and Wadim Zudilin, Further explorations of Boyd's conjectures and a conductor 21 elliptic curve, J. Lond. Math. Soc. (2) 93 (2016), no. 2, 341-360.
• 32. Wayne M. Lawton, A problem of Boyd concerning geometric means of polynomials, J. Number Theory 16 (1983), no. 3, 356-362.
• 33. D. H. Lehmer, Factorization of certain cyclotomic functions, Ann. of Math. (2) 34 (1933), no. 3, 461-479.
• 34. Douglas Lind, Klaus Schmidt, and Tom Ward, Mahler measure and entropy for commuting automorphisms of compact groups, Invent. Math. 101 (1990), no. 3, 593-629.
• 35. Anton Mellit, Elliptic dilogarithms and parallel lines, J. Number Theory 204 (2019), 1-24.
• 36. Michael J. Mossinghoff, Polynomials with small Mahler measure, Math. Comp. 67 (1998), no. 224, 1697-1705, S11-S14.
• 37. Michael J. Mossinghoff, Georges Rhin, and Qiang Wu, Minimal Mahler measures, Experiment. Math. 17 (2008), no. 4, 451-458.
• 38. Fernando Rodriguez Villegas, Modular Mahler measures. I, Ahlgren, Scott D. (ed.) et al., Topics in number theory (University Park, PA, 1997), Math. Appl., vol. 467, Kluwer Acad. Publ., Dordrecht, 1999, pp. 17-48.
• 39. Mathew Rogers and Wadim Zudilin, From $L$-series of elliptic curves to Mahler measures, Compos. Math. 148 (2012), no. 2, 385-414.
• 40. Mathew Rogers and Wadim Zudilin, On the Mahler measure of $1+X+1/X+Y+1/Y$, Int. Math. Res. Not. IMRN (2014), no. 9, 2305-2326.
• 41. Carl Ludwig Siegel, Algebraic integers whose conjugates lie in the unit circle, Duke Math. J. 11 (1944), 597-602.
• 42. Chris J. Smyth, On the product of the conjugates outside the unit circle of an algebraic integer, Bull. London Math. Soc. 3 (1971), 169-175.
• 43. Chris J. Smyth, On measures of polynomials in several variables, Bull. Aust. Math. Soc. 23 (1981), no. 1, 49-63.
• 44. Chris J. Smyth, The Mahler measure of algebraic numbers: a survey, McKee, James (ed.) et al., Number theory and polynomials, London Math. Soc. Lecture Note Ser., vol. 352, Cambridge Univ. Press, Cambridge, 2008, pp. 322-349.
• 45. Chris J. Smyth, Seventy years of Salem numbers, Bull. Lond. Math. Soc. 47 (2015), no. 3, 379-395.

### Affiliation

Boyd, David W.
Department of Mathematics, Univ. of British Columbia, Vancouver, B.C. V6T 1Z2, Canada