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

Abstract

Let $A_{1},\dotsc,A_{r}$ be linear partial differential operators in $N$ variables, with constant coefficients in a field $\mathbb{K}$ of characteristic $0$. With $\mathbf{A}:=(A_{1},\dotsc,A_{r})$, a polynomial $u$ is $\mathbf{A}$-harmonic if $\mathbf{A}u=0$, that is, $A_{1}u=\dotsb =A_{r}u=0$.

Denote by $m_{i}$ the order of the first nonzero homogeneous part of $A_{i}$ (initial part). The main result of this paper is that if $r\leq N$, the dimension over $\mathbb{K}$ of the space of $\mathbf{A}$-harmonic polynomials of degree at most $d$ is given by an explicit formula depending only upon $r$, $N$, $d$, and $m_{1},\dotsc,m_{r}$ (but not $\mathbb{K}$) provided that the initial parts of $A_{1},\dotsc,A_{r}$ satisfy a simple generic condition. If $r>N$ and $ A_{1},\dotsc,A_{r}$ 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 $A_{1},\dotsc,A_{r}$ have infinite order, which is unambiguous since they act only on polynomials. This is used to prove, as a corollary, the same formula when $A_{1},\dotsc,A_{r}$ are replaced with finite difference operators. Another application, when $\mathbb{K}=\mathbb{C}$ and $A_{1},\dotsc,A_{r}$ have finite order, yields dimension formulas for spaces of $\mathbf{A}$-harmonic polynomial-exponentials.