Mahler measure

Mahler measure for single variable polynomial

def (Mahler measure)

For \(P(x)=a \prod_{j=1}^{d} (x-\alpha_j) \in \mathbb{C}[x]\), define \[ M(P)=|a|\prod_{j=1}^d \max(|\alpha_j|,1) \]

looking for big primes

• \(P(x)=\prod_{i} (x-\alpha_i) \in \mathbb{Z}[x]\) be monic
• for each \(n\geq 1\), let \(\Delta_n=\prod_{i}(\alpha_i^n-1)\)
• find primes among the factors of \(\Delta_n\) as factoring \(\Delta_n\) is much easier than factoring a random number of the same size
• as \(\Delta_m|\Delta_n\) if \(m|n\), it is enough to consider \(\Delta_p/\Delta_1\)
• \(\Delta_n\) grows like \(M(P)^n\)

\[ \lim_{n\to \infty} \frac{|\alpha^{n+1}-1|}{|\alpha^{n}-1|} = \begin{cases} |\alpha|\, \mbox{ if } |\alpha|> 1, \\ 1\, \mbox{ if }|\alpha|<1 \end{cases} \]

• it is natural to consider polynomials \(P\) with a small value of \(M(P)\)

examples

• \(m(x^3+x+1)=0.382245085840\cdots\)
• \(m(x^3-x-1)=0.28119957432\cdots\)

Lehmer's question

Question

Can \(m(P)\) be arbtrarily small but positive for \(P(x)\in \mathbb{Z}[x]\)?

• The following is the smallest known positive value of \(m(P)\)

\[m(x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1)=0.1623576120\cdots\] \[M(x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1)=1.1762808182599175\cdots\]

Mahler's multivariate generalization

logarithmic Mahler measure

• We also define the logarithmic Mahler measure \(m(p):=\log M(P)\)
• one can compute \(m(P)\) by Jensen's formula

\[ \frac{1}{2\pi i}\int_{|x|=1} \log|P(x)| \; \frac{dx}{x} = \sum_{\alpha} \log^{+} |\alpha| \] where \(\log^{+} |\alpha|=\log \max(|\alpha|,1)\)

• Jensen's formula

\[ \int_{0}^{1}\log |e^{2\pi i \theta}-\alpha|\, \;d\theta=\log^{+}|\alpha| \]

multivariate logarithmic Mahler measure

• 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

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

• no explicit formula is known for polynomials in several variables

formula of Smyth

thm [Smyth1981]

\[ m(1+x_1+x_2)=L_{-3}'(-1)=\frac{3\sqrt{3}}{4\pi}L_{-3}(2)=0.3230659472\cdots \label{Smyth1} \]

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

• Smyth formula for Mahler measures

Multivariate Mahler measure

• Boyd conjecture on Mahler measure of three variables polynomial
• Mahler measure and L-values of elliptic curves
• Mahler measure, hyperbolic geometry and dilogarithm
• Rodriguez-Villegas conjecture on linear Mahler measures

related items

• Zagier's conjecture
• A log tangent integral
• Talk on introduction to Mahler measure

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

• 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 http://arxiv.org/pdf/math/0701397.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

• van Ittersum, Jan-Willem. “An Equivariant Version of Lehmer’s Conjecture on Heights.” arXiv:1602.01749 [math], February 4, 2016. http://arxiv.org/abs/1602.01749.
• Abdalaoui, El Houcein El. “Combinatorial Analysis of Polynomial Flatness Problem, Mahler Measure and Ergodic Theory.” arXiv:1508.06439 [math], August 26, 2015. http://arxiv.org/abs/1508.06439.
• Defant, Andreas, and Mieczysław Mastyło. “\(L^p\)-Norms and Mahler’s Measure of Polynomials on the \(n\)-Dimensional Torus.” arXiv:1508.05556 [math], August 22, 2015. http://arxiv.org/abs/1508.05556.
• Samuels, Charles L. “Continued Fraction Expansions in Connection with the Metric Mahler Measure.” arXiv:1508.01726 [math], August 7, 2015. http://arxiv.org/abs/1508.01726.
• 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
• 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.
• [Smyth1981] 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.

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• [{'LOWER': 'mahler'}, {'LEMMA': 'measure'}]