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$\alpha $-analytic function; polyanalytic function; zero set; Radó's theorem
Let $\Omega \subset \mathbb {C}^n$ be a bounded, simply connected $\mathbb C$-convex domain. Let $\alpha \in \mathbb Z_+^n$ and let $f$ be a function on $\Omega $ which is separately $C^{2\alpha _j-1}$-smooth with respect to $z_j$ (by which we mean jointly $C^{2 \alpha _j-1}$-smooth with respect to $\mathop {\rm Re} z_j$, $ \mathop {\rm Im} z_j$). If $f$ is $\alpha $-analytic on $\Omega \setminus f^{-1}(0)$, then $f$ is $\alpha $-analytic on $\Omega $. The result is well-known for the case $\alpha _i=1$, $1\leq i\leq n$, even when $f$ a priori is only known to be continuous.
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