I am aware that in general finding solutions of system of linear Diophantine equations is difficult and theoretically an open problem. (Please correct me if I am wrong.)
How about for the special case of homogenous system of linear Diophantine equations?
$$a_1x_1+\dots+a_nx_n=0\\ b_1x_1+\dots+b_nx_n=0\\ c_1x_1+\dots+c_nx_n=0\\ \dots$$
where we are looking for integer solutions?
Clearly $x_1=\dots=x_n=0$ is the trivial solution.
Do we know when there exists nontrivial solutions?
Thanks. (I am still interested in partial / weak results if the full result is unknown.)