| 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 Fröberg 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. |
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