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 what would be translated from German as Dirichlet's Drawers principle, which is widely known as the Pigeonhole principle.
[3] Suppose we are given a system of M linear equations in N unknowns such that N > M, say where the coefficients are integers, not all 0, and bounded by B.
The system then has a solution with the Xs all integers, not all 0, and bounded by Bombieri & Vaaler (1983) gave the following sharper bound for the X's: where D is the greatest common divisor of the M × M minors of the matrix A, and AT is its transpose.
Their proof involved replacing the pigeonhole principle by techniques from the geometry of numbers.