Rational functions with a polynomial iterate (Q1911636)

Let \(K\) be an algebraically closed field of characteristic \(p\). A description is given of rational functions \(f\in K(X)\), some iterate of which is a polynomial. It has been known that if \(K\) is the complex number field, then either \(f(X)\) of \(f(f(X))\) must be a polynomial. The author shows that the same is true in the case of characteristic \(p\) provided \(f(X)\) is separable and proves that in the inseparable case such functions must be linearly conjugate to \((az^q+ b)/(cz_q+ d)\) with \(q\) being a power of \(p\) and \(ad- bc\neq 0\).