Let p be a complex valued polynomial of two ral variables: $$ \sum {a_{ij} x^i y^j } $$ write: $$p(z)= \sum {P_j \overline z } ^j $$ where each $P_j$ is of the form $ P_j = \sum {b_{ij} z^i } $ Prove that p is an entire function if and only iff $ P_j \equiv 0 $
Clearly I have to consider the "derivate" $ P_{\overline z } = \frac{1} {2}\left( {P_x + iP_y } \right) $ , since the real and imaginary part are $C^1$ functions, being holomorphic it's equivalent of satisfy C.R , or equivalently $ P_{\overline z } = 0 $ in this case $$ P_{\overline z } = \sum {jP_j \overline z ^{j - 1} } = 0 $$ and now what can I do?