Maybe it is not a standard term, a harmonic polynomial is $h_{n,m}(z,\bar{z})=f_n(z)+\overline{g_m(z)}$ where $f$ and $g$ are polynomials in $z\in\mathbb{C}$ of degrees $n,m$ respectively.We may assume that $n\geq m$.
Then how many roots of the equation $h_{n,m}(z,\bar{z})=0$ are there?
I heard a saying that it has at least $n$ roots using $\textit{Argument Principle}$.
Is it right?I do not know how to apply this theorem.I have a test on $z^2-\bar{z}=0$,$z^2-\bar{z}+1=0$.In these cases the conclusion holds.But I do not know how to get the general result.
Will someone be kind enough to give me some hints on this problem?And will someone be kind enough to show me how to solve such equations on $\textit{Mathematica}$?
Thank you very much!