Mahler measure

introduction

• for a Laurent polynomial $P(x_1,\cdots, x_n)\in \mathbb{Z}[x_1^{\pm 1},\cdots,x_n^{\pm 1}]$, the (logarithmic) Mahler measure is defined to be

$$ \begin{aligned} m(P):&=\int_{0}^{1}\cdots \int_{0}^{1} \ln |P(e^{2\pi i \theta_1},\cdots, e^{2\pi i \theta_n})|\, d\theta_1\cdots d\theta_n\\ &= \frac{1}{(2\pi i)^n}\int_{|x_1|=\dots=|x_n|=1} \log|P(x_1,\dots,x_n)| \; \frac{dx_1}{x_1} \dots \frac{dx_n}{x_n} \end{aligned} $$

monic polynomial

• For a monic polynomial in one variable $P \in \mathbb{C}[x]$ one can compute $m(P)$ by Jensen's formula

$$ \frac{1}{2\pi i}\int_{|x|=1} \log|P(x)| \; \frac{dx}{x} = \sum_{\alpha:P(\alpha)=0} \max(0,\log|\alpha|)\,, $$

• but no explicit formula is known for polynomials in several variables.

example

• $m(1 + x - x^3 - x^4 - x^5 - x^6 - x^7 + x^9 + x^{10})=0.1623576120\cdots$
• $m(x^3-x-1)=0.28119957432\cdots$
• $m(x^3+x+1)=0.382245085840\cdots$

Multivariate Mahler measure

Smyth

thm [Smith1981]

$$ m(1+x_1+x_2)=L_{-3}'(-1)=\frac{3\sqrt{3}}{4\pi}L_{-3}(2)=0.3230659472\cdots $$

$$ m(1+x_1+x_2+x_3)=14\zeta'(-2)=\frac{7}{2\pi^2}\zeta(3)=0.4262783988\cdots $$

Rodriguez-Villegas

conjecture

$$ m(1+x_1+x_2+x_3+x_4)=-L_{f}'(-1)=\frac{675\sqrt{15}}{16\pi^5}L_{f}(4)=0.5444125617\cdots $$

• Rodriguez-Villegas conjecture on linear Mahler measures

$$ m(1+x_1+x_2+x_3+x_4+x_5)=-8L_{g}'(-1)=\frac{648}{\pi^6}L_{g}(5)=0.6273170748\cdots $$ where $$ f(\tau)=\eta(3\tau)^3\eta(5\tau)^3+\eta(\tau)^3\eta(15\tau)^3 $$ and $$ g(\tau)=\eta(\tau)^2\eta(2\tau)^2\eta(3\tau)^2\eta(6\tau)^2 $$

Deninger

conjecture

$$ m(1+x+\frac{1}{x}+y+\frac{1}{y})=L_{15A}'(0)=\frac{15}{4\pi^2}L_{15A}(2)=0.2513304337\cdots $$

Boyd

conjecture

$$ m(1+x+\frac{1}{x}+y+\frac{1}{y}+xy+\frac{1}{xy})=L_{14A}'(0)=\frac{7}{2\pi^2}L_{14A}(2)=0.2274812230\cdots $$

$$ m(-1+x+\frac{1}{x}+y+\frac{1}{y}+xy+\frac{1}{xy})=L_{30A}'(0)=\frac{15}{2\pi^2}L_{30A}(2)=0.6168709387\cdots $$

$$ m(3+x+\frac{1}{x}+y+\frac{1}{y})=\frac{21}{2\pi^2}L(E_{21},2) $$

$$ m(12+x+\frac{1}{x}+y+\frac{1}{y})=\frac{24}{\pi^2}L(E_{48},2) $$

• Boyd conjecture on Mahler measure of three variables polynomial
• Mahler measures and L-values of elliptic curves

related items

• Zagier's conjecture
• A log tangent integral

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxSTAxSnVRUmpEeFU/edit
• http://mathworld.wolfram.com/LehmersMahlerMeasureProblem.html
• http://mathworld.wolfram.com/MahlerMeasure.html

encyclopedia

• http://en.wikipedia.org/wiki/Mahler_measure

expositions

• Sengun, Mehmet Haluk. 2014. “An Introduction to A-Polynomials and Their Mahler Measures.” arXiv:1401.7484 [math]. http://arxiv.org/abs/1401.7484.
• Bertin, Marie-José, and MATILDE LALÍN. Mahler Measure of Multivariable Polynomials Women in Numbers 2: Research Directions in Number Theory 606 (2013): 125.
• Smyth, Chris. 2008. “The Mahler Measure of Algebraic Numbers: a Survey.” In Number Theory and Polynomials, 352:322–349. London Math. Soc. Lecture Note Ser. Cambridge: Cambridge Univ. Press. http://www.maths.ed.ac.uk/~chris/Smyth240707.pdf
• Lalin Mahler measures as values of regulators 2006
• Finch, Modular Forms on $SL_2(\mathbb{Z})$ 2005
• The many aspects of Mahler's measure, Banff workshop, 2003
  • final report
• Lalin Introduction to Mahler measure, 2003
• Boyd, Mahler's measure, hyperbolic geometry and the dilogarithm 2002

lecture notes

• Course at Harvard University Spring 2002.
• Fernando Rodriguez Villegashttp://www.ma.utexas.edu/users/villegas/KL/
• Suggested exercises on periods.
• The following are class notes taken by Sam Vandervelde (samv@mandelbrot.org).
• Notes 1
• Notes 2
• Notes 3
• Notes 4
• Notes 5
• The following are class notes taken by Matilde Lalin (mlalin@math.harvard.edu).
• Notes 6

articles

• Cochrane, Todd, R. M. S. Dissanayake, Nicholas Donohoue, M. I. M. Ishak, Vincent Pigno, Chris Pinner, and Craig Spencer. ‘Minimal Mahler Measure in Real Quadratic Fields’. arXiv:1410.4482 [math], 16 October 2014. http://arxiv.org/abs/1410.4482.
• Erdelyi, Tamas. “The Mahler Measure of the Rudin-Shapiro Polynomials.” arXiv:1406.2233 [math], June 9, 2014. http://arxiv.org/abs/1406.2233.
• Zudilin, Wadim. 2013. “Regulator of Modular Units and Mahler Measures”. ArXiv e-print 1304.3869. http://arxiv.org/abs/1304.3869.
• A dynamical interpretation of the global canonical height on an elliptic curve
• Mahler's Measure and the Dilogarithm (I)
• Mahler's Measure and the Dilogarithm (II)
  • Authors: David W. Boyd, Fernando Rodriguez-Villegas, Nathan M. Dunfield
• C. J. Smyth, An explicit formula for the Mahler measure of a family of 3-variable polynomials, J. Th. Nombres Bordeaux 14 (2002), 683{700
• Boyd, David W. 1998. “Mahler’s Measure and Special Values of $L$-Functions.” Experimental Mathematics 7 (1): 37–82.
• Deninger, Christopher. Deligne periods of mixed motives, K-theory and the entropy of certain Zn-actions Journal of the American Mathematical Society 10.2 (1997): 259-282.
• [Smith1981] Smyth, C. J. 1981. “On Measures of Polynomials in Several Variables.” Bulletin of the Australian Mathematical Society 23 (1): 49–63. doi:http://dx.doi.org/10.1017/S0004972700006894.

links

• http://enyokoyama.blogspot.com/2012/05/mahlers-measure-hyperbolic-geometry-and.html