On Polynomial Solutions of Linear Partial Differential and (q-)Difference Equations

We prove that the question of whether a given linear partial differential or difference equation with polynomial coefficients has non-zero polynomial solutions is algorithmically undecidable. However, for equations with constant coefficients this question can be decided very easily since such an equation has a non-zero polynomial solution iff its constant term is zero. We give a simple combinatorial proof of the fact that in this case the equation has polynomial solutions of all degrees. For linear partial q-difference equations with polynomial coefficients, the question of decidability of existence of non-zero polynomial solutions remains open. Nevertheless, for such equations with constant coefficients we show that the space of polynomial solutions can be described algorithmically. We present examples which demonstrate that, in contrast with the differential and difference cases where the dimension of this space is either infinite or zero, in the q-difference case it can also be finite and non-zero.