# 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.

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.
• "$S$ is injective and therefore has degree one." Why $S$ must be degree one? For example, $S(z)=1/z$, $S^{-1}=1/z$ is rational. Nov 1, 2022 at 14:52
• @hbghlyj: $S(z) = 1/z$ is injective and has degree one. Nov 1, 2022 at 14:53
• You are right. The definition of degree of $\frac{p(x)}{q(x)}$ where $\gcd(p(x),q(x))=1$ is $\max(\deg p,\deg q)$. (Related: math.stackexchange.com/questions/3389265) Thanks! Nov 1, 2022 at 15:01