The Khavinson-Shapiro conjecture and polynomial decompositions

The main result of the paper states the following: Let ψ be a polynomial in n variables. Suppose that there exists a constant C>0 such that any polynomial f has a polynomial decomposition f=ψqf+hf with Δkhf=0 and degqf⩽degf+C. Then degψ⩽2k. Here Δk is the kth iterate of the Laplace operator Δ. As an application, new classes of domains in Rn are identified for which the Khavinson-Shapiro conjecture holds.