New properties of harmonic polynomials (Q1316856)

Let \(H^ t_ n\) be the linear space of forms of degree \(t\); \(\text{Harm}^ t_{m,n}\) be the linear subspace in \(H^ t_ n\) consisting of those homogeneous polyharmonic polynomials \(\phi\) of order \(m\) and degree \(t\) that satisfy the polyharmonic equation \(D_{\sigma^ m}(\phi)= 0\). The main result (theorem 2) gives a dual description of polyharmonic polynomials, namely, \(\phi\in H^ t_ n\) is equivalent to the assertions \(\phi\in \text{Harm}^ t_{m,n}\) and \(D_ \phi(\sigma^ p)= 0\) for the same range of \(p\). To prove this result it is established (theorem 3) for \(\phi\in \text{Harm}^ t_{m,n}\) (harmonic polynomials) the representation of \(D_ \phi(\sigma^{t+p})\) via \(\sigma^ p\phi\), which is of independent interest. An application of the main result is also given (theorem 4).