This is part of the exercise 5.10 from the book "basic linear partial differential equations" by Treves:
" Let $P(z)$ be a polynomial in one variable, with complex coefficients. Describe all solutions in $\mathbb{R^2}$ of the partial differential equation $P(\frac{\partial}{\partial \bar{z}})h=0$. "
I tried a lot of methods end up being messed up. For example, it is natural to write $h=u+iv$ and $P(z)=R(z)+iT(z)$, where u,v are real valued functions and R, T are polynomials with real coefficients. Therefore we have $R(\frac{\partial}{\partial \bar{z}})u-T(\frac{\partial}{\partial \bar{z}})v=0$ , and $T(\frac{\partial}{\partial \bar{z}})u+R(\frac{\partial}{\partial \bar{z}})v=0$. But unfortunately, I got stuck here. I also tried using change of variables, but I found it was impossible to derive the desired form. I also carefully read the textbook, but I didn't get any hint. Actually, I don't feel good in doing this question, because I don't have much background in solving specific ODEs.
Could anyone give me some help? I would really appreciate it.
I'll try to take a stab at this. Analogously to the situation with ODEs, suppose $P(z)$ is a polynomial with distinct roots $r_1, \dots, r_n$. Then $$e^{r_1 \bar{z}}, \dots, e^{r_n \bar{z}}$$ should each be solutions to the equation $P(\frac{\partial}{\partial \bar{z}})h=0$. Indeed, they would make up a fundamental solution set, and the general solution should be: $$h(z,\bar{z}) = h_1(z)e^{r_1 \bar{z}} + \cdots+ h_n(z)e^{r_1 \bar{z}},$$ where $h_1, \dots, h_n$ are entire functions.
If we have roots of $P(z)$ with multiplicity, I suppose we must modify our fundamental solutions in the usual way by multiplying by powers of $\bar{z}$.