Sonia Ayers

2022-07-13

How is called the class of functions whose inverse function is a polynomial? Is there any study of such functions?

Zackery Harvey

The functions you're interested in are a particular subset of the algebraic functions. To be specific, an algebraic function $f\left(x\right)$ is a function that satisfies
${a}_{n}\left(x\right)f\left(x{\right)}^{n}+{a}_{n-1}\left(x\right)f\left(x{\right)}^{n-1}+\cdots +{a}_{0}\left(x\right)=0$
where the ${a}_{i}\left(x\right)$ are polynomials. By contrast, let
$P\left(x\right)={b}_{n}{x}^{n}+{b}_{n-1}{x}^{n-1}+\cdots +{b}_{0}$
be an arbitrary polynomial. Its inverse, ${P}^{-1}\left(x\right)$, satisfies
$P\left({P}^{-1}\left(x\right)\right)={b}_{n}{P}^{-1}\left(x{\right)}^{n}+{b}_{n-1}{P}^{-1}\left(x{\right)}^{n-1}+\cdots +{b}_{0}=x.$
So inverses of polynomials are those algebraic functions for which the defining coefficient functions ${a}_{i}$ are constants, except for the last, which is a constant minus $x$.

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