Siegel's lemma

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.


Wikimedia Foundation. 2010.

Игры ⚽ Поможем сделать НИР

Look at other dictionaries:

  • Carl Ludwig Siegel — Infobox Scientist name = Carl Ludwig Siegel image width = 242 x 360 22k caption = Carl Ludwig Siegel birth date = birth date|1896|12|31 birth place = Berlin, Germany death date = death date and age|1981|4|4|1896|12|31 death place = Göttingen,… …   Wikipedia

  • Sidney Siegel — (4 January 1916, New York 29 November 1961) was an American psychologist who became especially well known for his work in popularising non parametric statistics for use in the behavioural sciences. He was a co developer of the statistical test… …   Wikipedia

  • List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… …   Wikipedia

  • List of lemmas — This following is a list of lemmas (or, lemmata , i.e. minor theorems, or sometimes intermediate technical results factored out of proofs). See also list of axioms, list of theorems and list of conjectures. 0 to 9 *0/1 Sorting Lemma ( comparison… …   Wikipedia

  • Auxiliary function — In mathematics, auxiliary functions are an important construction in transcendental number theory. They are functions which appear in most proofs in this area of mathematics and that have specific, desirable properties, such as taking the value… …   Wikipedia

  • Transcendence theory — In mathematics, transcendence theory is a branch of number theory that investigates transcendental numbers, in both qualitative and quantitative ways.TranscendenceThe fundamental theorem of algebra tells us that if we have a non zero polynomial… …   Wikipedia

  • Pigeonhole principle — A photograph of pigeons in holes. Here there are n = 10 pigeons in m = 9 holes, so by the pigeonhole principle, at least one hole has more than one pigeon: in this case, both of the top corner holes contain two pigeons. The principle says nothing …   Wikipedia

  • Liste mathematischer Sätze — Inhaltsverzeichnis A B C D E F G H I J K L M N O P Q R S T U V W X Y Z A Satz von Abel Ruffini: eine allgemeine Polynomgleichung vom …   Deutsch Wikipedia

  • Poaceae —   Poáceas …   Wikipedia Español

  • Parity of zero — Zero objects, divided into two equal groups Zero is an even number. In other words, its parity the quality of an integer being even or odd is even. Zero fits the definition of even number : it is an integer multiple of 2, namely 0 × 2. As a… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”