"Mahler measure"의 두 판 사이의 차이

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

2014년 2월 16일 (일) 17:36 판

introduction

  • for a Laurent polynomial $P(x_1,\cdots, x_n)\in \mathbb{Z}[x_1^{\pm 1},\cdots,x_n^{\pm 1}]$, the 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.


examples

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

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

$$ m(1+x_1+x_2+x_3+x_4+x_5)=-8L_{g}'(-1)=\frac{648}{\pi^6}L_{g}(5)=0.6273170748\cdots $$ 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

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



Boyd

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

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


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