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Let $f:ℝ→ℝ$ be a real analytic function with infinitely many isolated zeros. Let us define the function: $$h(s₁,s₂,...,s_{r+1})=\prod_{k=1}^{r+1}f^{(k+1)}(\left(1-2\prod_{j=1}^{k}s_{j}\right)$$ Also, all the $k$-th derivatives of $f$ have infinitely many isolated zeros. Does $h$ have infinitely many isolated zeros? I know that $f′(1-2s₁)=0$ has infinitely many isolated zeros.

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Maybe f was real-valued, not vector valued? –  Pietro Majer Apr 10 '13 at 18:39
    
@Pietro Majer: I changed the function to be $f:ℝ→ℝ$. –  China-Hong Kong Apr 10 '13 at 18:56

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up vote 2 down vote accepted

I assume $f$ was a real-valued analytic function on $(0,1)$, otherwise I do not understand the notation for $h$. But then no zero of $h$ can be isolated. Indeed, the zero set of $h$ is the union of the zero sets of its $r+1$ factors, and all of them vanish on some $r$ dimensional submanifold of $(0,1)^{r+1}$.

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