TITLE: is there a solution to this linear Diophantine system?
QUESTION [6 upvotes]: I have a matrix $A \in \mathbb Z^{n \times m}$, where $m > n$, and a vector $b \in \mathbb Z^n$. Under what conditions does
$$Ax = b$$
have an integer solution? Is there a way to bound the norm of the solution vector $x$ in terms of the norms of $A$ and $b$?
Essentially, I want something like Siegel's lemma, but for the non-homogeneous case.
I am not an expert on this and will appreciate any help. Thanks!

REPLY [7 votes]: This kind of questions arise very often in integer linear optimization. It is well-known that the bitsize of a solution will be polynomially bounded by the sizes of of $A$ and $b$, i.e. by the maximum bitsize of entries of $A$ and $b$, and by $\max(n,m)$. See e.g. Corollary 5.2a in the A.Schrijver's book. 
There are many sufficient conditions known for the existence of an integer solution, e.g. $A$ being totally unimodular (i.e. each square submatrix has determinant 0,1,or -1) is one.
Your problem is in fact easier (in optimization one often assumes $x\geq 0$), and Will gives a good suggestion in his comment above.
The book in Sect. 5.3 also gives more details on the algorithm Igor described in his answer.