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Let $P(x,y)$ be a polynomial in two variables that is not identically $0$. Let $f:U\to\mathbb{C}$ be a holomorphic function defined on a region $U\subset \mathbb{C}$ such that $P(\Re(f(z)),\Im(f(z)))=0$ for all $z\in U$. Show that $f$ is constant.

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