Given a multivariate compactly supported distribution \(\phi\), we derive here a necessary and sufficient condition for the global linear independence of its integer translates. This condition is based on the location of the zeros of \({\hat \phi}=the\) Fourier-Laplace transform of \(\phi\). The utility of the condition is demonstrated by several examples and applications, showing, in particular, that previous results on box splines and exponential box splines can be derived from this condition by a simple combinatorial argument.