Given $r$ vectors $v_1, \dots, v_r$ in $\mathbb{Z}^n$, is there an easy way (in terms of the entries of the $v_i$) to determine if there is a point of $\mathbb{Z}^n$ in the interior of the simplex spanned by the $v_i$? I'm especially thinking of the case where $r$ is much smaller than $n$. Of course one can assume the $v_i$ are independent.
For example, in case $n = 2$ and $r = 1$, one is effectively asking whether the greatest common divisor of the (two) entries of $v_1$ is nonone. When $r = n$, one can look at the determinant of the vectors.