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 = ψ q f + h f with Δ k h f = 0 and deg q f ⩽ deg f + C . Then deg ψ ⩽ 2 k . Here Δ k is the kth iterate of the Laplace operator Δ. As an application, new classes of domains in R n are identified for which the Khavinson–Shapiro conjecture holds.