Suppose that $X$ is an affine real algebraic variety. In particular $X$ is the zero locus of a set of $k$ degree-two polynomials in $\mathbb{R}^N$. (If it makes any difference, we also happen to know that $X$ is compact in the subspace topology.)

Let $F : X \to \mathbb{R}$ be projection on to one of the coordinates. My understanding is that even if $F$ is not flat and proper, the set of $r \in \mathbb{R}$ such that $F$ is not smooth at $r$ is still constructible, and for a one-dimensional variety such as $\mathbb{R}$ the Zariski topology is the cofinite topology, so any constructible set that is not the whole space must be finite.

What I would like is to make this effective.

Can we give an upper bound on the number of critical values of $F$, in terms of $k$ and $N$?

For example, is there some way to apply Milnor-Thom?

Suppose that $X$ is an affine real algebraic variety. In particular $X$ is the zero locus of a set of $k$ degree-two polynomials in $\mathbb{R}^N$. (If it makes any difference, we also happen to know that $X$ is compact in the subspace topology.)

Let $F : X \to \mathbb{R}$ be projection on to one of the coordinates. My understanding is that even if $F$ is not flat and proper, the set of $r \in \mathbb{R}$ such that $F$ is not smooth at $r$ is still constructible, and for a one-dimensional variety such as $\mathbb{R}$ the Zariski topology is the cofinite topology, so any constructible set that is not the whole space must be finite.

What I would like is to make this effective.

Can we give an upper bound on the number of critical values of $F$, in terms of $k$ and $N$?

For example, is there some way to apply Milnor-Thom?

Added: I do not need an exact answer, but only asymptotic upper bounds. For example if $k \approx N$ is it true that the number of critical values of $F$ is bounded by $e^{cN}$ for some constant $c$?