What do we know about homogenous system of linear Diophantine equations?

Question

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.)

Answer 1

Yes, there is a "theory" of solving systems of linear Diophantine equations, see for example the article Linear Diophantine Equations, based on the Extended Euclidean Algorithm. For further references see

Algorithm for finding integer solutions to a system of linear Diophantine equations