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| In [[transcendence theory|transcendental]] [[number theory]] and [[Diophantine approximation]], '''Siegel's lemma''' refers to bounds on the solutions of linear equations obtained by the construction of [[auxiliary function]]s. The existence of these polynomials was proven by [[Axel Thue]];<ref>
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| {{cite journal|last = Thue|first = Axel|authorlink = Axel Thue|title = Über Annäiherungswerte algebraischer Zahlen|journal = [[Crelle's Journal|J. Reine Angew. Math.]]|volume=135|year = 1909|pages = 284–305|ref = harv}}</ref> Thue's proof used [[Dirichlet's box principle]]. [[Carl Ludwig Siegel]] published his lemma in 1929.<ref>{{cite journal|last = Siegel|first = Carl Ludwig|authorlink = Carl Ludwig Siegel|title = Über einige Anwendungen diophantischer Approximationen|journal = Abh. Pruess. Akad. Wiss. Phys. Math. Kl.|year = 1929|pages = 41–69|ref = harv}}, reprinted in Gesammelte Abhandlungen, volume 1; the lemma is stated on page 213</ref> It is a pure [[existence theorem]] for a [[system of linear equations]].
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| Siegel's lemma has been refined in recent years to produce sharper bounds on the estimates given by the lemma.<ref>
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| {{cite journal|last = Bombieri|first = E.|authorlink = Enrico Bombieri|coauthors = Mueller, J.|title = On effective measures of irrationality for <math>{\scriptscriptstyle\sqrt[r]{a/b}}</math> and related numbers|journal = Journal für die reine und angewandte Mathematik|volume = 342|year = 1983|pages = 173–196}}</ref>
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| ==Statement==
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| Suppose we are given a system of ''M'' linear equations in ''N'' unknowns such that ''N'' > ''M'', say
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| :<math>a_{11} X_1 + \cdots+ a_{1N} X_N = 0</math>
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| :<math>\cdots</math>
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| :<math>a_{M1} X_1 +\cdots+ a_{MN} X_N = 0</math>
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| where the coefficients are rational integers, not all 0, and bounded by ''B''. The system then has a solution
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| :<math>(X_1, X_2, \dots, X_N)</math>
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| with the ''X''s all rational integers, not all 0, and bounded by
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| :<math>(NB)^{M/(N-M)}.\,</math><ref>{{harv|Hindry|Silverman|2000}} Lemma D.4.1, page 316.</ref>
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| {{harvtxt|Bombieri|Vaaler|1983}} gave the following sharper bound for the ''X'''s:
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| :<math>\max|X_j|\le \left(D^{-1}\sqrt{\det(AA^T)}\right)^{1/(N-M)}</math>
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| where ''D'' is the greatest common divisor of the ''M'' by ''M'' minors of the matrix ''A'', and ''A''<sup>''T''</sup> is its transpose.
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| Their proof involved replacing the [[Dirichlet box principle]] by techniques from the [[geometry of numbers]].
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| ==See also==
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| *[[Diophantine approximation]]
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| ==References==
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| {{reflist}}
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| *{{Cite journal|last = Bombieri|first = E.|last2= Vaaler|first2= J.|title = On Siegel's lemma|journal = Inventiones Mathematicae|volume = 73|issue = 1|year=1983|pages = 11–32|url = http://www.springerlink.com/content/k55042224131lp42|doi = 10.1007/BF01393823|ref = harv|postscript = <!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}
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| *{{Cite book | last1=Hindry | first1=Marc | author1-link=Marc Hindry | last2=Silverman | first2=Joseph H. | author2-link=Joseph H. Silverman | title=Diophantine geometry | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics | isbn=978-0-387-98975-4; 978-0-387-98981-5 | mr=1745599 | year=2000 | volume=201 | ref=harv | postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}
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| * [[Wolfgang M. Schmidt]]. ''Diophantine approximation''. Lecture Notes in Mathematics 785. Springer. (1980 [1996 with minor corrections]) (Pages 125-128 and 283-285)
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| * Wolfgang M. Schmidt. "Chapter I: Siegel's Lemma and Heights" (pages 1–33). ''Diophantine approximations and Diophantine equations'', Lecture Notes in Mathematics, Springer Verlag 2000.
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| [[Category:Lemmas]]
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| [[Category:Diophantine approximation]]
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| [[Category:Diophantine geometry]]
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