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Throughout, $F_q$ will denote a finite field of $q$ elements with characteristic $p \neq 2$.

It is well-known that the equation $y^2 = f(x)$ (for square-free $f \in F_q[X]$) defines an hyper-elliptic function field. It means that $F_q(x)(\sqrt{f(x)})\cap \overline{F_q} = F_q$.

What about the general equation $g(y) = f(x)$ ($\deg g > 1$) ? Does it satisfy this property too, under some conditions?

There are 3 special cases that interest me:

  1. $g(y)$ being a single monomial $y^{2k}$
  2. $ \deg g = \deg f$
  3. The intersection of 1 and 2

I know of some counter examples, satisfying $g(x) \equiv f(x) + C$ for some $C \in F_q$ and $g' \equiv D \in F_q$.

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