The main result of the paper states the following: Let ψ be a polynomial
in n variables of degree t: Suppose that there exists a constant C > 0 such
that any polynomial f has a polynomial decomposition f = ψ qf + hf with
khf = 0 and deg qf deg f + C: Then deg ψ 2k. Here ∆k is the kth
iterate of the Laplace operator ∆ : As an application, new classes of domains
in Rn are identi ed for which the Khavinson-Shapiro conjecture holds.
