Faltings’s Version of Siegel’s Lemma

Abstract

Siegel’s Lemma in its original form guarantees the existence of a small non-trivial integral solution of a system of linear equations with rational integer coefficients and with more variables than equations. It reads as follows: 1 Lemma. (C.L. Siegel) Let A = (a ij ) be an N × M matrix with rational integer coefficients. Put a = max i,j |a ij |. Tien, if N < M, the equation Ax = 0 has a solution x ∈ ℤ M , x≠0, with $$ \left\| x \right\| \leqslant (Ma)^{N/(M - N)} $$ where ‖ ‖ denotes the max-norm: ‖x‖=‖(x1,...,x M ‖=max1≤i≤M|x i | in ℝ M .