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I have the following problem. Suppose

$$ f(z, \overline{z} )= \sum a_{lm} z^l \overline{z}^m$$

is a polynomial. ($z \in \mathbb{C}$). then $f$ contains $\mathbf{no} $ $\mathbf{term}$ with $m > 0$ ( $f$ contains no $\overline{z}$ terms) iff $\frac{ \partial f}{\partial \overline{z}} = 0 $

I really don't understand what the problem is asking for. In particular, What does it mean for $f$ to have no term with $m > 0$ ??

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"$f$ contains no term with $m > 0$" means that $a_{lm} = 0$ for all $m > 0$, i.e., $f$ is a polynomial in $z$ (and not just a polynomial in $z$ and $\bar{z}$). The problem is asking you to show that this is the case if and only if $\frac{\partial f}{\partial \bar{z}} = 0$.

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