Algebraic approximation in CR geometry

We prove the following CR version of Artin's approximation theorem for holomorphic mappings between real-algebraic sets in complex space. Let $M\subset \C^N$ be a real-algebraic CR submanifold whose CR orbits are all of the same dimension. Then for every point $p\in M$, for every real-algebraic subset $S'\subset \C^N\times\C^{N'}$ and every positive integer $\ell$, if $f\colon (\C^N,p)\to \C^{N'}$ is a germ of a holomorphic map such that ${\rm Graph}\, f \cap (M\times \C^{N'})\subset S'$, then there exists a germ of a complex-algebraic map $f^\ell \colon (\C^N,p)\to \C^{N'}$ such that ${\rm Graph}\, f^\ell \cap (M\times \C^{N'})\subset S'$ and that agrees with $f$ at $p$ up to order $\ell$.

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