Question regarding pluriharmonic function

Question

A real valued function $f$ defined on an open subset $U$ of $\mathbb{C}^n$ is said to be Pluriharmonic if $$\frac{\partial^2}{\partial z_i\partial\bar{z_j}}f\equiv0,$$ for $1\leq i,j \leq n.$ I was reading some paper, in which it was used that any Pluriharmonic function can be written in the form of $f=Re P +r$, where $P$ is a homoegeneous holomorphic polynomial of degree k, k$\geq$1, and $r=o(|z|^k)$. Can anyone help me in proving this. I know that any pluriharmonic funtion can be written as Real part of a holomorphic function. Can this fact help in proving this?

Answer 1

First of all, you can only assert that pluriharmonic functions locally can be written as the real part of a holomorphic function. Furhermore there seems to something missing in your decomposition: the right hand side vanishes at $z=0$.

But, if you're happy with a local result, and assume that $f(0) = 0$ (or add a constant to the right hand side), then $f = \operatorname{Re} g$, and if you do a Maclaurin expansion of $g$, you can take $P$ as the homogeneous polynomial corresponding to the lowest order terms in the Maclaurin expansion, and $r = \operatorname{Re}(g-P)$.