lets assume we have a real vectorspace $V$ and functions $f_1, \dots, f_k \colon V \to \mathbb{R}$ which are real-analytic (for instance, let them be polynomial). Furthermore we have an embedded real-analytic submanifold $M \subset V$ and a point $x \in M$, such that $$F := (f_1|_M, \dots, f_k|_M) \colon M \to \mathbb{R}^k$$ is a submersion in $x$, i.e. $$d_xF \colon T_xM \to T_{F(x)} \mathbb{R}^k$$ is surjective. Do we know something about the set of points $M_0 = \{ y \in M \ | \ F \text{ is a submersion in } y\}$?

Since $M$ is an embedded submanifold, $M_0$ is open in $M$. But is it dense?

Edit: Thanks to the comments of Willie Wong and Holonomia my question is answered, if I know, that $f_j|_M \colon M \to \mathbb{R}$ are real analytic. So if $y \in M$ is a point, we find a real analytic "submanifold" chart $(U,\phi)$, with $U$ open in $V$, s.th. $f \circ \phi^{-1} \colon \phi(U) \to \mathbb{R}$ is real analytic and such that $\phi(U\cap M) = \phi(U) \cap (\mathbb{R}^k \times \{0\})$. With the subspace-topology in $\mathbb{R}^n$, we can now conclude, that at each point $z \in \phi(U \cap M)$ it is locally (in $\phi(U\cap M)$) given as a convergent power series. Therefore $f|_M \colon M \to \mathbb{R}$ is real analytic.

Is this right? Do we find such "submanifold" charts?

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