Cauchy, Goursat and Dirichlet problems for holomorphic partial differential equations

n this paper we survey recent results about Fischer decomposi-
tions of polynomials or entire functions and their applications to holomorphic
partial di erential equations. We discuss Cauchy and Goursat problems for the
polyharmonic operator. Special emphasis is given to the Khavinson-Shapiro
conjecture concerning polynomial solvability of the Dirichlet problem.