View source for Transcendency, measure of

''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

<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>

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:

<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>

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

<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>

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" />,

<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>

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]]]).

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



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.B. Shidlovskii,   "Transcendental numbers" , de Gruyter  (1989)  (Translated from Russian)</TD></TR></table>