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Concerning the definition of a class of functions introduced by Nilsson

In the paper 'Some growth and ramification properties of certain integrals on algebraic manifolds' by Nilsson we find the following definitions: My question is how does one prove the remark "It ...
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Complex analytic function $f$ on $\mathbb{C}^n$ vanish on real sphere must vanish on complex sphere

I'm considering a complex entire function $f$: $\mathbb{C}^n\to \mathbb{C}$. Suppose $f=0$ on $\{(x_1,\cdots,x_n)\in\mathbb{R}^n:\sum_{k=1}^n x_k^2=1\}$. I want to prove $$f=0\textit{ on } M_1=\{(z_1,\...