The generic dimension of spaces of $\mathbf{A}$-harmonic polynomials

Abstract

Let A1,...,Ar be linear partial differential operators in N variables, with constant coefficients in a field K of characteristic 0. With A := (A1,...,Ar), a polynomial u is A-harmonic if Au = 0, that is, A1u = ··· = Aru = 0. Denote by mi the order of the first nonzero homogeneous part of Ai (initial part). The main result of this paper is that if r ≤ N, the dimension over K of the space of A-harmonic polynomials of degree at most d is given by an explicit formula depending only upon r, N, d, and m1,...,mr (but not K) provided that the initial parts of A1,...,Ar satisfy a simple generic condition. If r > N and A1,...,Ar are homogeneous, the existence of a generic formula is closely related to a conjecture of Froberg on Hilbert functions. The main result holds even if A1,...,Ar have infinite order, which is unambiguous since they act only on polynomials. This is used to prove, as a corollary, the same formula when A1,...,Ar are replaced with finite difference operators. Another application, when K = C and A1,...,Ar have finite order, yields dimension formulas for spaces of A-harmonic polynomial-exponentials.