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

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    $\begingroup$ Algebraic functions. Although algebraic can be more than just inverses, but implicit functions defined by polynomial equations. en.wikipedia.org/wiki/Algebraic_function $\endgroup$ – user119256 Jan 8 '14 at 0:25
  • $\begingroup$ Every polynomial is a function, but not every polynomial has an inverse function. $\endgroup$ – hardmath Jan 8 '14 at 0:28
  • $\begingroup$ @hardmath In the same way that saying a "polynomial is a function" is not strictly correct (one must specify a domain an a codomain to give a function) one can talk about inverses of functions without inverses by restricting the domain when necessary. Why be picky if you are also going to use loose terminology? $\endgroup$ – user119256 Jan 8 '14 at 0:31
  • $\begingroup$ @Karene I doubt that inverse to any algebraic function is a polynomial. $\endgroup$ – Anixx Jan 8 '14 at 2:42
  • $\begingroup$ @Anixx Your doubt is correct. As I said already, algebraic functions are functions defined by polynomial equations (implicit function defined). Being inverse is a particular case of implicitly define function. $\endgroup$ – user119256 Jan 8 '14 at 3:11
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The functions you're interested in are a particular subset of the algebraic functions. To be specific, an algebraic function $f(x)$ is a function that satisfies $$ a_n(x)f(x)^n+a_{n-1}(x)f(x)^{n-1}+\cdots+a_0(x)=0 $$ where the $a_i(x)$ are polynomials. By contrast, let $$P(x)=b_n x^n + b_{n-1}x^{n-1} + \cdots + b_0$$ be an arbitrary polynomial. Its inverse, $P^{-1}(x)$, satisfies $$ P(P^{-1}(x))=b_nP^{-1}(x)^n+b_{n-1}P^{-1}(x)^{n-1}+\cdots+b_0=x. $$ 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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Unless we want a restriction that the inverse be "globally" a polynomial, i.e. the inverse of a monotone polynomial to ensure the requirement of being a function, the phrase "inverse polynomial" will probably convey the intended class of functions well enough.

Interpolation by inverse polynomials, esp. inverse quadratic interpolation, has been studied in connection with root-finding and line searches in optimization.

For degree 1 the inverse polynomial doesn't yield any new methods. For degree 2 the inverse (quadratic) polynomial has found a popular niche in Brent's algorithm.

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