Suppose I am solving a set of equations (over $\mathbb{Z}$) of the form $Av=0$. Assuming that a non-zero solution exists, how large can the smallest solution be?
More precisely, let $A$ be a matrix over integers such that the entries in $A$ are bounded above by some integer $M$. Does there exist $v \in N(A) \setminus \{0\}$ such that each entry of $v$ is atmost $M^{O(1)}$? (Here $N(A)$ denotes null space of $A$ and assume that $N(A) \setminus \{0\}$ is non-empty).
Over $\mathbb{R}$, I could have easily said: $Av=0 \implies A\left(\frac{v}{||v||}\right) = 0$ and hence we can always find a small (norm 1) null vector, whose all entries are in $[-1,1]$. But I am not sure how do I achieve something like this over $\mathbb{Z}$.