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Recently, I read the relation between Shabat polynomials and trees. The book [0] says that if $p: \mathbb{C} \to \mathbb{C}$ is a degree-$n$ polynomial such that the segment $[-1,1]$ contains no critical value of $p$ then $p^{-1} ([-1,1])$ has $n$ connected components and each is homeomorphic to $[-1,1]$.

I can't argue why this should be true. Only things that I can say is that $p$ becomes a $n$-sheeted covering map from $p^{-1} ([-1,1])$ to $[-1,1]$. Any kind of help will be appreciated.


[0] Sergei K. Lando, Alexander K. Zvonkin, Graphs on Surfaces and their Applications. (Chapter 2: Dessins d’Enfants)

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  • $\begingroup$ Can you describe some $n$-sheeted covering map of $[-1,1]$ not of that prescribed form? $\endgroup$ – Gerald Edgar Apr 7 '18 at 21:51
  • $\begingroup$ Although only tangentially related, it may be worth looking into Reeb's theorem for spheres to see how this works for the case in which we have only $2$ critical values for our polynomial and how we can use an automorphism on $/mathbb{C}$ to set them on $0$ and $1$ so we can use Shabat's theorem. $\endgroup$ – Casquibaldo Apr 7 '18 at 23:08
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If $f$ has no critical values on $[-1,1]$ then the restriction of $$f:f^{-1}([-1,1])\to [-1,1]$$ is a covering map. This covering has degree $n=\deg f$, so there are $n$ inverse branches of $f$ on $[-1,1]$. The mages of $[-1,1]$ under these branches are $n$ distinct curves, components of $f^{-1}$. For elementary properties of covering maps see Ahlfors, Conformal invariants, or Massey, Algebraic Topology. An Introduction.

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