Kernels of discrete convolutions and subdivision operators

We consider kernels of discrete convolution operators or, equivalently, homogeneous solutions of partial difference operators and show that these solutions always have to be exponential polynomials. The respective polynomial space in connected directly though somewhat intricately to the multiplicity of the common zeros of certain multivariate polynomials, a concept introduced by Gr\"obner in the description of kernels of partial differential operators with constant coefficients. These results can are then used to determine the kernels of stationary subdivision operators as well.