In mathematics, specifically in

transcendental number theory Transcendental number theory is a branch of number theory that investigates transcendental numbers (numbers that are not solutions of any polynomial equation with rational coefficients), in both qualitative and quantitative ways. Transcendence ...

and

Diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated by ...

, Siegel's lemma refers to bounds on the solutions of

linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coeffici ...

s obtained by the construction of auxiliary functions. The existence of these

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...

s was proven by

Axel Thue Axel Thue (; 19 February 1863 – 7 March 1922) was a Norwegian mathematician, known for his original work in diophantine approximation and combinatorics. Work Thue published his first important paper in 1909. He stated in 1914 the so-called ...

; Thue's proof used

Dirichlet's box principle In mathematics, the pigeonhole principle states that if items are put into containers, with , then at least one container must contain more than one item. For example, if one has three gloves (and none is ambidextrous/reversible), then there mu ...

Carl Ludwig Siegel Carl Ludwig Siegel (31 December 1896 – 4 April 1981) was a German mathematician specialising in analytic number theory. He is known for, amongst other things, his contributions to the Thue–Siegel–Roth theorem in Diophantine approximation ...

published his lemma in 1929. It is a pure

existence theorem In mathematics, an existence theorem is a theorem which asserts the existence of a certain object. It might be a statement which begins with the phrase " there exist(s)", or it might be a universal statement whose last quantifier is existential ...

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.

Statement

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

a_ X_1 + \cdots+ a_ X_N = 0

\cdots

a_ X_1 +\cdots+ a_ X_N = 0

where the

coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...

s 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 :

(NB)^.

Lemma D.4.1, page 316. gave the following sharper bound for the ''Xs: :

\max, X_j,  \,\le \left(D^\sqrt\right)^

where ''D'' is the

greatest common divisor In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' i ...

of the ''M'' × ''M'' minors of the

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...

''A'', and ''A''^''T'' is its

transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The tr ...

. Their proof involved replacing the

pigeonhole principle In mathematics, the pigeonhole principle states that if items are put into containers, with , then at least one container must contain more than one item. For example, if one has three gloves (and none is ambidextrous/reversible), then there m ...

by techniques from the

geometry of numbers Geometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice in \mathbb R^n, and the study of these lattices provides fundamental informatio ...

References

* *{{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 Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 ...

, location=Berlin, New York , series=Graduate Texts in Mathematics , isbn=978-0-387-98981-5 , mr=1745599 , year=2000 , volume=201 * Wolfgang M. Schmidt. ''Diophantine approximation''. Lecture Notes in Mathematics 785. Springer. (1980 996 with minor corrections (Pages 125-128 and 283-285) * 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. Lemmas Diophantine approximation Diophantine geometry

Statement

See also

References