Siegel's lemma

In mathematics, specifically 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;^[1] Thue's proof used Dirichlet's box principle. Carl Ludwig Siegel published his lemma in 1929.^[2] 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.^[3]

Statement

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 integers, not all 0, and bounded by B. The system then has a solution

(X_{1},X_{2},\dots ,X_{N})

with the Xs all integers, not all 0, and bounded by

(NB)^{M/(N-M)}.

^[4]

Bombieri & Vaaler (1983) gave the following sharper bound for the X's:

\max |X_{j}|\,\leq \left(D^{-1}{\sqrt {\det(AA^{T})}}\right)^{\!1/(N-M)}

where D is the greatest common divisor of the M × M minors of the matrix A, and A^T is its transpose. Their proof involved replacing the pigeonhole principle by techniques from the geometry of numbers.

^ Thue, Axel (1909). "Über Annäherungswerte algebraischer Zahlen". J. Reine Angew. Math. 1909 (135): 284–305. doi:10.1515/crll.1909.135.284. S2CID 125903243.
^ Siegel, Carl Ludwig (1929). "Über einige Anwendungen diophantischer Approximationen". Abh. Preuss. Akad. Wiss. Phys. Math. Kl.: 41–69., reprinted in Gesammelte Abhandlungen, volume 1; the lemma is stated on page 213
^ Bombieri, E.; Mueller, J. (1983). "On effective measures of irrationality for ${\scriptscriptstyle {\sqrt[{r}]{a/b}}}$ and related numbers". Journal für die reine und angewandte Mathematik. 342: 173–196.
^ (Hindry & Silverman 2000) Lemma D.4.1, page 316.

Bombieri, E.; Vaaler, J. (1983). "On Siegel's lemma". Inventiones Mathematicae. 73 (1): 11–32. Bibcode:1983InMat..73...11B. doi:10.1007/BF01393823. S2CID 121274024.
Hindry, Marc; Silverman, Joseph H. (2000). Diophantine geometry. Graduate Texts in Mathematics. Vol. 201. Berlin, New York: Springer-Verlag. ISBN 978-0-387-98981-5. MR 1745599.
Wolfgang M. Schmidt. Diophantine approximation. Lecture Notes in Mathematics 785. Springer. (1980 [1996 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.