Which kinds of rational functions of one variable have an inverse relation that contains a branch that is a rational function?

Ausanioyck

Ausanioyck

Answered question

2022-02-15

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?

Answer & Explanation

Alissia Head

Alissia Head

Beginner2022-02-16Added 5 answers

If R is a rational function and S a branch of R1 in an open set UC 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.
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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