A characterization of the Khavinson-Shapiro conjecture via Fischer operators

The Khavinson-Shapiro conjecture states that ellipsoids are the only bounded domains in euclidean space satisfying the following property (KS): the solution of the Dirichlet problem for polynomial data is polynomial. In this paper we show that property (KS) for a domain $\mathrm{\Omega }$ is equivalent to the surjectivity of a Fischer operator associated to the domain $\mathrm{\Omega }\mathrm{.}$