Given any field \(K\) and a polynomial \(p\in K[X]= K[X_1,\dots,X_n]\), the differential operator \(p(D)\) on \(K[X]\) is defined by substituting \(\partial/\partial x_i\) for the variable \(X_i\). For the case that \(K\) is algebraically closed for characteristic 0, and \(p\) is homogeneous, the set of homogeneous solutions of the PDE \(p(D)=0\) is determined. A \(d\)-form \(q\) is a solution if and only if \(q\) belongs to the homogeneous part of degree \(d\) in the ideal of \(K[X]\) generated by certain powers of certain linear forms. The method of proof is elementary (linear algebra and Hilbert's Nullstellensatz). Extensive historical notes firmly embed the paper in its classical background.