"Mahler measure"의 두 판 사이의 차이

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==introduction==
 
==introduction==
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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
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$$
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m(P):=\int_{0}^{1}\cdots \int_{0}^{1} \ln |P(e^{2\pi i \theta_1},\cdots, e^{2\pi i \theta_n})|\, d\theta_1\cdots d\theta_n
 +
$$
  
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==examples==
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===Smyth===
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$$
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m(1+x_1+x_2)=L_{-3}'(-1)=\frac{3\sqrt{3}}{4\pi}L_{-3}(2)=0.3230659472\cdots
 +
$$
 +
 +
$$
 +
m(1+x_1+x_2+x_3)=14\zeta'(-2)=\frac{7}{2\pi^2}\zeta(3)=0.4262783988\cdots
 +
$$
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 +
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===Rodriguez-Villegas===
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$$
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m(1+x_1+x_2+x_3+x_4)=-L_{f}'(-1)=\frac{675\sqrt{15}}{16\pi^5}L_{f}(4)=0.5444125617\cdots
 +
$$
 +
 +
$$
 +
m(1+x_1+x_2+x_3+x_4+x_5)=-8L_{g}'(-1)=\frac{648}{\pi^6}L_{g}(5)=0.6273170748\cdots
 +
$$
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where
 +
$$
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f(\tau)=\eta(3\tau)^3\eta(5\tau)^3+\eta(\tau)^3\eta(15\tau)^3
 +
$$
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and
 +
$$
 +
g(\tau)=\eta(\tau)^2\eta(2\tau)^2\eta(3\tau)^2\eta(6\tau)^2
 +
$$
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 +
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===Deninger===
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$$
 +
m(1+x+\frac{1}{x}+y+\frac{1}{y})=L_{15A}'(0)=\frac{15}{4\pi^2}L_{15A}(2)=0.2513304337\cdots
 +
$$
 +
 +
 +
 +
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===Boyd===
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$$
 +
m(1+x+\frac{1}{x}+y+\frac{1}{y}+xy+\frac{1}{xy})=L_{14A}'(0)=\frac{7}{2\pi^2}L_{14A}(2)=0.2274812230\cdots
 +
$$
  
  
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==expositions==
 
==expositions==
 +
* 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
 
* Matilde N. Laln [http://www.math.ualberta.ca/%7Emlalin/ubc.pdf Mahler measures as values of regulators] 2006
 
* Matilde N. Laln [http://www.math.ualberta.ca/%7Emlalin/ubc.pdf Mahler measures as values of regulators] 2006
 
* Finch, [http://www.people.fas.harvard.edu/~sfinch/csolve/frs.pdf Modular Forms on $SL_2(\mathbb{Z})$] 2005
 
* Finch, [http://www.people.fas.harvard.edu/~sfinch/csolve/frs.pdf Modular Forms on $SL_2(\mathbb{Z})$] 2005

2013년 8월 27일 (화) 07:32 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

$$ m(P):=\int_{0}^{1}\cdots \int_{0}^{1} \ln |P(e^{2\pi i \theta_1},\cdots, e^{2\pi i \theta_n})|\, d\theta_1\cdots d\theta_n $$


examples

Smyth

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

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


Rodriguez-Villegas

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

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

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


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