On the existence of spatially tempered null solutions to linear constant coefficient PDES

Given a linear, constant coefficient partial differential equation in ℝd+1, where one independent variable plays the role of ‘time’, a distributional solution is called a null solution if its past is zero. Motivated by physical considerations, distributional solutions that are tempered in the spatial directions alone (with no restriction in the time direction) are considered. An algebraic-geometric characterization is given, in terms of the polynomial describing the PDE, for the null solution space to be trivial (that is, consisting only of the zero distribution).