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''transcendence measure''
 
''transcendence measure''
  
A function characterizing the deviation of a given [[Transcendental number|transcendental number]] from a set of algebraic numbers of bounded degree and bounded height under a change of bounds on these parameters. For a transcendental number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093590/t0935901.png" /> and natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093590/t0935902.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093590/t0935903.png" />, the measure of transcendency is
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A function characterizing the deviation of a given [[Transcendental number|transcendental number]] from a set of algebraic numbers of bounded degree and bounded height under a change of bounds on these parameters. For a transcendental number $\omega$ and natural numbers $n$ and $H$, the measure of transcendency is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093590/t0935904.png" /></td> </tr></table>
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$$w_n(\omega;H)=\min|P(\omega)|,$$
  
where the minimum is taken over all non-zero integer polynomials of degree not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093590/t0935905.png" /> and height not exceeding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093590/t0935906.png" />. It follows from Dirichlet's  "box"  principle (cf. [[Dirichlet principle|Dirichlet principle]]) that the following always holds:
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where the minimum is taken over all non-zero integer polynomials of degree not exceeding $n$ and height not exceeding $H$. It follows from Dirichlet's  "box"  principle (cf. [[Dirichlet principle|Dirichlet principle]]) that the following always holds:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093590/t0935907.png" /></td> </tr></table>
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$$w_n(\omega;H)<c_1^nH^{-n},$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093590/t0935908.png" /> depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093590/t0935909.png" />. In many cases it is possible to obtain not only a proof of the transcendency of a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093590/t09359010.png" /> but also a lower bound for the measure of transcendency in terms of the degree, and logarithmic or exponential functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093590/t09359011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093590/t09359012.png" />. For example, Hermite's method of proof of transcendency of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093590/t09359013.png" /> enables one to obtain the inequality
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where $c_1$ depends only on $\omega$. In many cases it is possible to obtain not only a proof of the transcendency of a number $\omega$ but also a lower bound for the measure of transcendency in terms of the degree, and logarithmic or exponential functions of $n$ and $H$. For example, Hermite's method of proof of transcendency of $e$ enables one to obtain the inequality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093590/t09359014.png" /></td> </tr></table>
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$$w_n(e;h)>H^{-n-(c_2n^2\ln n)/\ln\ln H},$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093590/t09359015.png" /> is an absolute constant and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093590/t09359016.png" />. For any fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093590/t09359017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093590/t09359018.png" />,
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where $c_2>0$ is an absolute constant and $H\geq H_0(n)$. For any fixed $n$ and $\epsilon>0$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093590/t09359019.png" /></td> </tr></table>
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$$w_n(\omega;H)>c_3H^{-n-\epsilon},\quad c_3=c_3(\omega;n,\epsilon)$$
  
for almost-all (in the sense of Lebesgue) real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093590/t09359020.png" /> (see [[Mahler problem|Mahler problem]]). Transcendental numbers can be classified on the basis of the difference in asymptotic behaviour of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093590/t09359021.png" /> under unrestricted variation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093590/t09359022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093590/t09359023.png" /> (see [[#References|[3]]]).
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for almost-all (in the sense of Lebesgue) real numbers $\omega$ (see [[Mahler problem|Mahler problem]]). Transcendental numbers can be classified on the basis of the difference in asymptotic behaviour of $w_n(\omega;H)$ under unrestricted variation of $n$ and $H$ (see [[#References|[3]]]).
  
 
====References====
 
====References====

Revision as of 14:54, 2 August 2014

transcendence measure

A function characterizing the deviation of a given transcendental number from a set of algebraic numbers of bounded degree and bounded height under a change of bounds on these parameters. For a transcendental number $\omega$ and natural numbers $n$ and $H$, the measure of transcendency is

$$w_n(\omega;H)=\min|P(\omega)|,$$

where the minimum is taken over all non-zero integer polynomials of degree not exceeding $n$ and height not exceeding $H$. It follows from Dirichlet's "box" principle (cf. Dirichlet principle) that the following always holds:

$$w_n(\omega;H)<c_1^nH^{-n},$$

where $c_1$ depends only on $\omega$. In many cases it is possible to obtain not only a proof of the transcendency of a number $\omega$ but also a lower bound for the measure of transcendency in terms of the degree, and logarithmic or exponential functions of $n$ and $H$. For example, Hermite's method of proof of transcendency of $e$ enables one to obtain the inequality

$$w_n(e;h)>H^{-n-(c_2n^2\ln n)/\ln\ln H},$$

where $c_2>0$ is an absolute constant and $H\geq H_0(n)$. For any fixed $n$ and $\epsilon>0$,

$$w_n(\omega;H)>c_3H^{-n-\epsilon},\quad c_3=c_3(\omega;n,\epsilon)$$

for almost-all (in the sense of Lebesgue) real numbers $\omega$ (see Mahler problem). Transcendental numbers can be classified on the basis of the difference in asymptotic behaviour of $w_n(\omega;H)$ under unrestricted variation of $n$ and $H$ (see [3]).

References

[1] A.O. Gel'fond, "Transcendental and algebraic numbers" , Dover, reprint (1960) (Translated from Russian)
[2] P.L. Cijsouw, "Transcendence measures" , Univ. Amsterdam (1972) (Dissertation)
[3] A. Baker, "Transcendental number theory" , Cambridge Univ. Press (1975)


Comments

References

[a1] A.B. Shidlovskii, "Transcendental numbers" , de Gruyter (1989) (Translated from Russian)
How to Cite This Entry:
Transcendency, measure of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transcendency,_measure_of&oldid=32681
This article was adapted from an original article by V.G. Sprindzhuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article