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Difference between revisions of "Algebraic independence, measure of"

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The measure of algebraic independence of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011540/a0115401.png" /> is the function
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The measure of algebraic independence of the numbers $\alpha_1,\dots,\alpha_m$ is the function
  
<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/a/a011/a011540/a0115402.png" /></td> </tr></table>
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$$\Phi(\alpha_1,\dots,\alpha_m;n,H)=\min|P(\alpha_1,\dots,\alpha_m)|,$$
  
where the minimum is taken over all polynomials of degree at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011540/a0115403.png" />, with rational integer coefficients not all of which are zero, and of height at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011540/a0115404.png" />. For more details see [[Transcendency, measure of|Transcendency, measure of]].
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where the minimum is taken over all polynomials of degree at most $n$, with rational integer coefficients not all of which are zero, and of height at most $H$. For more details see [[Transcendency, measure of|Transcendency, measure of]].

Revision as of 13:23, 14 September 2014

The measure of algebraic independence of the numbers $\alpha_1,\dots,\alpha_m$ is the function

$$\Phi(\alpha_1,\dots,\alpha_m;n,H)=\min|P(\alpha_1,\dots,\alpha_m)|,$$

where the minimum is taken over all polynomials of degree at most $n$, with rational integer coefficients not all of which are zero, and of height at most $H$. For more details see Transcendency, measure of.

How to Cite This Entry:
Algebraic independence, measure of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraic_independence,_measure_of&oldid=33284
This article was adapted from an original article by A.B. Shidlovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article