Difference between revisions of "Transcendency, measure of"
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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 | + | 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 |
| − | + | $$w_n(\omega;H)=\min|P(\omega)|,$$ | |
| − | where the minimum is taken over all non-zero integer polynomials of degree not exceeding | + | 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: |
| − | + | $$w_n(\omega;H)<c_1^nH^{-n},$$ | |
| − | where | + | 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 | + | 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 | + | 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
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