Let $M_m := (f_1, \cdots, f_m )$ be an algebraic map from $\mathbb{R}^n$ to $\mathbb{R}^m$ and $f_1^2,...,f_m^2$ are homogeneous polynomials of the same degree in $Q[x_1,...,x_n]$ . Similarly define $M_{m-1} := (f_1, \cdots, f_{m-1} )$ such that $M_{m-1}(\mathbb{R}^n) \subseteq \mathbb{R}^{m-1}$. So there is a natural projection $\Pi$ from $M_m(\mathbb{R}^n)$ to $M_{m-1}(\mathbb{R}^n)$.
Let $P_{m-1}$ be a proper linear subspace in $\mathbb{R}^{m-1}$. Define a generic point $p$ in $P_{m-1} \cap M_{m-1}(\mathbb{R}^n)$ as follows:
We say $p$ is a generic point in $P_{m-1} \cap M_{m-1}(\mathbb{R}^n)$ if there exists a neighbourhood $N(p) \subseteq P_{m-1}\cap M_{m-1}(\mathbb{R}^n)$ such that $\dim(N(p)) = \dim(P_{m-1})$. (we assume $\dim(M_{m-1}(\mathbb{R}^n)) \geq \dim(P_{m-1})$.
My question is: can I conclude that the dimension of the fibre of a generic point $p$ (the fibre is the preimage $\Pi^{-1}(p)$ in $M_{m}(\mathbb{R}^n)$) is the same for every such generic point $p$?
