For $P$ a non constant polynomial, if $[a, b]$ contains no critical points of $P$, can we determine $P^{-1}([a, b])$?

Question

Let be $P \in \mathbb{C}[X]$ a non constant polynomial.

I'll use the notation $P^{-1}$ for the preimage.

If $[a, b]$ ($a, b \in \mathbb{R}$) contains no critical points of $P$, can we have information on $P^{-1}([a, b])$ more than:

• it is closed by continuity ;
• it is bounded (because $\lim_{\lvert z\vert \to +\infty} P(z) = \pm \infty$)
• then, it's compact ;

My intuition is: it's some reunion related to the polynomial $P$, but unsure.