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1. In this note I present an algebraic method for constructing basic sets of polynomials which are solutions of a linear homogeneous partial differential equation with constant coefficients. This method generalizes and unifies several known results (see ?3). Let E=Rn (n>1) be the euclidean space of dimension n, whose points shall be x=(x1, * * *, xn). The capital letters M and J will denote multi-indices M= (m1, * * *, mn), J= (j * * * , jn), where the mi and ji are positive integers; the corresponding lower-case letters will mean m=1 M =ml+ +mn. We shall also write XJ= 1n We consider a linear homogeneous partial differential operator with

Basic sets of polynomial solutions for partial differential equations.