Let $P_N:\mathbb{R}^2\to \mathbb{R}$ be a homogeneous harmonic polynomial of degree $N (N\geq1)$. Prove the nodal lines $\{x\in\mathbb{R}:P_N(x)=0\}$ divide $\mathbb{R}^2$ into $2N$ regions.
I just encountered a problem regarding the nodal line of homogeneous harmonic polynomials. I understand that the dimension of homogeneous harmonic polynomial in the problem is 2, whose proof can be found on MSE How to calculate dimesion of homogeneous harmonic polynomial. Is this related to my problem? Also, I know there are some papers dealing the similar issue in the general case, but they seem too advanced for me. Could anyone write a detailed answer or suggest some basic reference?
Appreciate any help!