I have a matrix A \in Z^{n \by m}, where m > n and a vector b \in Z^n. Then, under what conditions does an integer solution exist to the equation

Ax = b.

Is there a way to bound the norm of the solution vector x in terms of norms of A and b?

Essentially I want something like Siegel's lemma but for the inhomogeneous case.

I am not an expert on this and will appreciate any help.

Thanks!