Siegel's lemma

In transcendental number theory and Diophantine approximation, Siegel's lemma refers to bounds on the solutions of linear equations obtained by the construction of auxiliary functions. The existence of these polynomials was proven by Axel Thue [cite journal|last = Thue|first = Axel|authorlink = Axel Thue|title = Über Annäiherungswerte algebraischer Zahlen|journal = J. Reine Angew. Math.|journallink=Crelle|volume=135|date = 1909|pages = 284-305] ; Thue's proof used Dirichlet's box principle. Carl Ludwig Siegel published his lemma in 1929 [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.|date = 1929|pages = 41-69] . It is a pure existence theorem for a system of linear equations.

Siegel's lemma has been refined in recent years to produce sharper bounds on the estimates given by the lemma. [cite journal|last = Bombieri|first = E.|authorlink = Enrico Bombieri|coauthors = Mueller, J.|title = On effective measures of irrationality for $\{scriptscriptstylesqrt\; [r]\; \{a/b$ and related numbers|journal = Journal für die reine und angewandte Mathematik|volume = 342|date = 1983|pages = 173-196]

tatement

Suppose we are given a system of "M" linear equations in "N" unknowns such that "N" > "M", say

:$a\_\{11\}\; X\_1\; +\; cdots+\; a\_\{1N\}\; X\_N\; =\; 0$

:$cdots$

:$a\_\{M1\}\; X\_1\; +cdots+\; a\_\{MN\}\; X\_N\; =\; 0$

where the coefficients are rational integers, not all 0, and bounded by "B". The system then has a solution

:$(X\_1,\; X\_2,\; dots,\; X\_N)$

with the "X"s all rational integers, not all 0, and bounded by

:$1\; +\; (NB)^\{M/(N-M)\}.,$ [cite journal|last = Bombieri|first = E.|coauthors = Vaaler, J.|title = On Siegel's lemma|journal = Inventiones Mathematicae|volume = 73|issue = 1|date = Feb 1983|pages = 11–32|url = http://www.springerlink.com/content/k55042224131lp42|doi = 10.1007/BF01393823]

ee also

*Diophantine approximation

References

* M. Hindry and J.H. Silverman, "Diophantine geometry", Springer Verlag, 2000, ISBN 0-387-98981-1. Page 316.