In this article, a positive answer is given to the following question posed by Hayman [35, page 326]: if a polyharmonic entire function of order vanishes on distinct ellipsoids in the Euclidean space , then it vanishes everywhere. Moreover, a characterization of ellipsoids is given in terms of an extension property of solutions of entire data functions for the Dirichlet problem, answering a question of Khavinson and Shapiro [39, page 460]. These results are consequences from a more general result in the context of direct sum decompositions (Fischer decompositions) of polynomials or functions in the algebra of all real-analytic functions defined on the ball of radius and center zero whose Taylor series of homogeneous polynomials converges compactly in . The main result states that for a given elliptic polynomial of degree and for sufficiently large radius , the following decomposition holds: for each function , there exist unique such that and . Another application of this result is the existence of polynomial solutions of the polyharmonic equation for polynomial data on certain classes of algebraic hypersurfaces