SCHWEITZER 2021 Problem 1

Let $n, m \in \mathbb{N}$ ; $a_1,\ldots, a_m \in \mathbb{Z}^n$ . Show that nonnegative integer linear combinations of these vectors give exactly the whole $\mathbb{Z}^n$ lattice, if $m \ge n$ and the following two statements are satisfied:

• The vectors do not fall into the half-space of $\mathbb{R}^n$ containing the origin (i.e. they do not fall on the same side of an $n-1$ dimensional subspace),
• the largest common divisor (not pairwise, but together) of $n \times n$ minor determinants of the matrix $(a_1,\ldots, a_m)$ (which is of size $m \times n$ and the $i$ -th column is $a_i$ as a column vector) is $1$ .