Mahler measure

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imported>Pythagoras0님의 2015년 1월 24일 (토) 23:32 판 (→‎Lehmer's conjecture)
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introduction

  • $P(x)=\prod_{i} (x-\alpha_i) \in \mathbb{Z}[x]$ 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 $\Delta_m|\Delta_n$ if $m|n$, it is enough to consider $\Delta_p/\Delta_1$


monic polynomial

def (Mahler measure)

For $P(x)=a \prod_{\alpha} (x-\alpha) \in \mathbb{C}[x]$, define $$ M(P(x))=|a|+\prod_{\alpha} \max(|\alpha|,1) $$ where the product is over all $\alpha$ satisfying $P(\alpha)=0$.

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

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

  • Jensen's formula

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

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

Lehmer's question

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

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

Mahler's multivariate generalization

  • 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

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


formula of Smyth

thm [Smyth1981]

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

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


two proofs of \ref{Smyth1}

  • direct calculation
  • using regulator

Multivariate Mahler measure


Rodriguez-Villegas

conjecture

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

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


related items

computational resource


encyclopedia


expositions


lecture notes

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
  • 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.