Which kinds of rational functions of one variable have an inverse relation that contains a branch that is a rational function?
Let's consider the rational functions whose numerator and denominator of the function term are coprime.
Which kinds of rational functions of one variable have an inverse relation that contains a branch that is a rational function?
Which kinds of polynomial functions of one variable have an inverse relation that contains a branch that is a rational function?
I assume the degree of the numerator and the degree of the denominator of the function term has to be less than or equal to $1$. Some calculations with algebraic equations with undetermined parameters as coefficients seem to show that. But I'm not sure.
$\endgroup$1 Answer
$\begingroup$If $R$ is a rational function and $S$ a branch of $R^{-1}$ in an open set $U \subset \Bbb C$ then $S(R(z)) = z$ in $U$. If $S$ is also a rational function then it follows that $S(R(z)) = z$ globally (as meromorphic functions).
It follows that $S$ is injective and therefore has degree one. Then $R$ has degree one as well.
So the only rational functions with a (local) rational branch of the inverse are rational functions of degree one (which are the Möbius transformations).
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