"Mahler measure"의 두 판 사이의 차이

수학노트
둘러보기로 가기 검색하러 가기
imported>Pythagoras0
 
(사용자 2명의 중간 판 60개는 보이지 않습니다)
1번째 줄: 1번째 줄:
−
==introduction==
+
==Mahler measure for single variable polynomial==
−
* 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
+
;def (Mahler measure)
−
$$
+
For <math>P(x)=a \prod_{j=1}^{d} (x-\alpha_j) \in \mathbb{C}[x]</math>, define
 +
:<math>
 +
M(P)=|a|\prod_{j=1}^d \max(|\alpha_j|,1)
 +
</math>
 +
 
 +
==looking for big primes==
 +
* <math>P(x)=\prod_{i} (x-\alpha_i) \in \mathbb{Z}[x]</math> be monic
 +
* for each <math>n\geq 1</math>, let <math>\Delta_n=\prod_{i}(\alpha_i^n-1)</math>
 +
* find primes among the factors of <math>\Delta_n</math> as factoring <math>\Delta_n</math> is much easier than factoring a random number of the same size
 +
* as <math>\Delta_m|\Delta_n</math> if <math>m|n</math>, it is enough to consider <math>\Delta_p/\Delta_1</math>
 +
* <math>\Delta_n</math> grows like <math>M(P)^n</math>
 +
:<math>
 +
\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}
 +
</math>
 +
* it is natural to consider polynomials <math>P</math> with a small value of <math>M(P)</math>
 +
====examples====
 +
* <math>m(x^3+x+1)=0.382245085840\cdots</math>
 +
* <math>m(x^3-x-1)=0.28119957432\cdots</math>
 +
 
 +
===Lehmer's question===
 +
;Question
 +
Can <math>m(P)</math> be arbtrarily small but positive for <math>P(x)\in \mathbb{Z}[x]</math>?
 +
 
 +
* The following is the smallest known positive value of <math>m(P)</math>
 +
:<math>m(x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1)=0.1623576120\cdots</math>
 +
:<math>M(x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1)=1.1762808182599175\cdots</math>
 +
 
 +
==Mahler's multivariate generalization==
 +
===logarithmic Mahler measure===
 +
* We also define the logarithmic Mahler measure <math>m(p):=\log M(P)</math>
 +
* one can compute <math>m(P)</math> by Jensen's formula
 +
:<math>
 +
\frac{1}{2\pi i}\int_{|x|=1} \log|P(x)| \; \frac{dx}{x} = \sum_{\alpha} \log^{+} |\alpha|
 +
</math>
 +
where <math>\log^{+} |\alpha|=\log \max(|\alpha|,1)</math>
 +
* Jensen's formula
 +
:<math>
 +
\int_{0}^{1}\log |e^{2\pi i \theta}-\alpha|\, \;d\theta=\log^{+}|\alpha|
 +
</math>
 +
 
 +
===multivariate logarithmic Mahler measure===
 +
* for a Laurent polynomial <math>P(x_1,\cdots, x_n)\in \mathbb{Z}[x_1^{\pm 1},\cdots,x_n^{\pm 1}]</math>, the (logarithmic) Mahler measure is defined to be
 +
:<math>
 
\begin{aligned}
 
\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\\
+
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{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}
 
\frac{dx_1}{x_1} \dots \frac{dx_n}{x_n}
 
\end{aligned}
 
\end{aligned}
−
$$
+
</math>
−
 
+
* no explicit formula is known for polynomials in several variables
−
 
 
−
==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.
 
  
 +
==formula of Smyth==
 +
;thm '''[Smyth1981]'''
 +
:<math>
 +
m(1+x_1+x_2)=L_{-3}'(-1)=\frac{3\sqrt{3}}{4\pi}L_{-3}(2)=0.3230659472\cdots \label{Smyth1}
 +
</math>
  
−
 
+
:<math>
−
==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
 
m(1+x_1+x_2+x_3)=14\zeta'(-2)=\frac{7}{2\pi^2}\zeta(3)=0.4262783988\cdots
−
$$
+
</math>
−
 
+
* [[Smyth formula for Mahler measures]]
−
 
 
−
===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]]
+
==Multivariate Mahler measure==
−
 
 
−
$$
 
−
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
 
−
$$
 
 
* [[Boyd conjecture on Mahler measure of three variables polynomial]]
 
* [[Boyd conjecture on Mahler measure of three variables polynomial]]
−
* [[Mahler measures and L-values of elliptic curves]]
+
* [[Mahler measure and L-values of elliptic curves]]
−
 
+
* [[Mahler measure, hyperbolic geometry and dilogarithm]]
 +
* [[Rodriguez-Villegas conjecture on linear Mahler measures]]
  
 
==related items==
 
==related items==
 
* [[Zagier's conjecture]]
 
* [[Zagier's conjecture]]
 
* [[A log tangent integral]]
 
* [[A log tangent integral]]
−
 
+
* [[Talk on introduction to Mahler measure]]
−
 
  
 
==computational resource==
 
==computational resource==
93번째 줄: 91번째 줄:
  
 
==expositions==
 
==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. [http://www.dms.umontreal.ca/~mlalin/surveyMahlerfinal-revised.pdf Mahler Measure of Multivariable Polynomials] Women in Numbers 2: Research Directions in Number Theory 606 (2013): 125.
 
* Bertin, Marie-José, and MATILDE LALÍN. [http://www.dms.umontreal.ca/~mlalin/surveyMahlerfinal-revised.pdf 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
+
* 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 [http://www.math.ualberta.ca/%7Emlalin/ubc.pdf Mahler measures as values of regulators] 2006
 
* Lalin [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 <math>SL_2(\mathbb{Z})</math>] 2005
 
* [http://www.birs.ca/workshops/2003/03w5035/ The many aspects of Mahler's measure], Banff workshop, 2003
 
* [http://www.birs.ca/workshops/2003/03w5035/ The many aspects of Mahler's measure], Banff workshop, 2003
 
** [http://www.birs.ca/workshops/2003/03w5035/report03w5035.pdf final report]
 
** [http://www.birs.ca/workshops/2003/03w5035/report03w5035.pdf final report]
 
* Lalin [http://www.dms.umontreal.ca/~mlalin/uba.pdf Introduction to Mahler measure], 2003
 
* Lalin [http://www.dms.umontreal.ca/~mlalin/uba.pdf Introduction to Mahler measure], 2003
 
* Boyd, [http://www.math.ca/notes/v34/n2/Notesv34n2.pdf Mahler's measure, hyperbolic geometry and the dilogarithm] 2002
 
* Boyd, [http://www.math.ca/notes/v34/n2/Notesv34n2.pdf Mahler's measure, hyperbolic geometry and the dilogarithm] 2002
−
* Boyd, David W. 1998. “Mahler’s Measure and Special Values of $L$-Functions.” Experimental Mathematics 7 (1): 37–82.
 
−
 
  
  
112번째 줄: 107번째 줄:
 
*  Suggested '''[http://www.ma.utexas.edu/users/villegas/KL/periods.dvi exercises]''' on periods.
 
*  Suggested '''[http://www.ma.utexas.edu/users/villegas/KL/periods.dvi exercises]''' on periods.
 
*  The following are class notes taken by Sam Vandervelde (samv@mandelbrot.org).
 
*  The following are class notes taken by Sam Vandervelde (samv@mandelbrot.org).
−
* [[3313085/attachments/2687571|sam-notes-1-1.pdf]]
 
−
* [[3313085/attachments/2687573|sam-notes-2.pdf]]
 
−
* [[3313085/attachments/2687575|sam-notes-3.pdf]]
 
−
* [[3313085/attachments/2687577|sam-notes-4.pdf]]
 
−
* [[3313085/attachments/2687579|sam-notes-5.pdf]]
 
 
* '''[http://www.ma.utexas.edu/users/villegas/KL/sam-notes-1.dvi Notes 1]'''
 
* '''[http://www.ma.utexas.edu/users/villegas/KL/sam-notes-1.dvi Notes 1]'''
 
* '''[http://www.ma.utexas.edu/users/villegas/KL/sam-notes-2.dvi Notes 2]'''
 
* '''[http://www.ma.utexas.edu/users/villegas/KL/sam-notes-2.dvi Notes 2]'''
124번째 줄: 114번째 줄:
 
*  The following are class notes taken by Matilde Lalin (mlalin@math.harvard.edu).
 
*  The following are class notes taken by Matilde Lalin (mlalin@math.harvard.edu).
 
* '''[http://www.ma.utexas.edu/users/villegas/KL/KL.ps Notes 6]'''
 
* '''[http://www.ma.utexas.edu/users/villegas/KL/KL.ps Notes 6]'''
−
 
−
 
  
 
==articles==
 
==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. “<math>L^p</math>-Norms and Mahler’s Measure of Polynomials on the <math>n</math>-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.
 
* 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
 
* A dynamical interpretation of the global canonical height on an elliptic curve
−
* [http://www.smc.math.ca/cjm/v54/p468 Mahler's Measure and the Dilogarithm (I)]
 
−
* [http://arxiv.org/abs/math.NT/0308041 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
 
* 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 <math>L</math>-Functions.” Experimental Mathematics 7 (1): 37–82.
 
* Deninger, Christopher. [http://www.mathaware.org/journals/jams/1997-10-02/S0894-0347-97-00228-2/S0894-0347-97-00228-2.pdf 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.
 
* Deninger, Christopher. [http://www.mathaware.org/journals/jams/1997-10-02/S0894-0347-97-00228-2/S0894-0347-97-00228-2.pdf 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.
+
* '''[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.
−
 
 
−
 
 
−
 
 
−
==links==
 
−
* http://enyokoyama.blogspot.com/2012/05/mahlers-measure-hyperbolic-geometry-and.html
 
  
  
 
[[분류:개인노트]]
 
[[분류:개인노트]]
−
[[Category:research topics]]
 
 
[[분류:dilogarithm]]
 
[[분류:dilogarithm]]
 +
[[분류:L-functions and L-values]]
 +
[[분류:migrate]]
 +
 +
==메타데이터==
 +
===위키데이터===
 +
* ID :  [https://www.wikidata.org/wiki/Q6734205 Q6734205]
 +
===Spacy 패턴 목록===
 +
* [{'LOWER': 'mahler'}, {'LEMMA': 'measure'}]

2021년 2월 17일 (수) 01:02 기준 최신판

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]

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

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

Multivariate Mahler measure

related items

computational resource


encyclopedia


expositions


lecture notes

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.

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'mahler'}, {'LEMMA': 'measure'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=Mahler_measure&oldid=51667"