Homogeneous Polynomial Solutions to Constant Coefficient PDE's

Abstract

Suppose p is a homogeneous polynomial over a field K. Let p(D) be the differential operator defined by replacing each occurrence of xj in p by xj , defined formally in case K is not a subset of C. (The classical example is p(x1 , ..., xn)= j xj , for which p(D) is the Laplacian { .) In this paper we solve the equation p(D) q=0 for homogeneous polynomials q over K, under the restriction that K be an algebraically closed field of characteristic 0. (This restriction is mild, considering that the ``natural'' field is C.) Our Main Theorem and its proof are the natural extensions of work by Sylvester, Clifford, Rosanes, Gundelfinger, Cartan, Maass and Helgason. The novelties in the presentation are the generalization of the base field and the removal of some restrictions on p. The proofs require only undergraduate mathematics and Hilbert's Nullstellensatz. A fringe benefit of the proof of the Main Theorem is an Expansion Theorem for homogeneous polynomials over R and C. This theorem is trivial for linear forms, goes back at least to Gauss for p=x1+x 2 2+x 2 3 , and has also been studied by Maass and Helgason.