Siegel's lemma

In transcendental number theory and Diophantine approximation, Siegel's lemma refers to the formalization of the construction of auxiliary polynomials. The existence of these polynomials was proven by Axel Thue in a 1909 paper entitled Über Annäiherungswerte algebraischer; Thue's proof used Dirichlet's box principle. Carl Ludwig Siegel published his lemma in 1929 in a paper titled Über einige Anwendungen diophantischer Approximationen. 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.^[1]

[edit] 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 rational 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 rational integers, not all 0, and bounded by

$1 + (NB)^{M/(N-M)}.\,$ ^[2]

^ 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.
^ Bombieri, E.; Vaaler, J. (Feb 1983). "On Siegel's lemma". Inventiones Mathematicae 73 (1): 11–32. doi:10.1007/BF01393823.