Fixed points of rational functions satisfying the Carlitz property (Q2280309)

A rational function \(f\) over a finite field \(\mathbb{F}_q\) is said to have the Carlitz property if there exists an \(a\in\mathbb{F}_q\) such that \[ f\left(\frac{xy+a}{x+y}\right) = \frac{f(x)f(y)+a}{f(x)+f(y)}. \] There are exactly three classes of functions satisfying the Carltz property, one of which is the class of Rédei functions [\textit{L. Carlitz}, Duke Math. J. 29, 325--332 (1962; Zbl 0196.31102)], the other two are referred to in this article as Type-1 and Type-3 functions. The authors investigate fixed points of functions satisfying thr Carlitz property. Explicitly they determine the fixed points for the Redei functions. For Type-1 and Type-3 functions, a method for algorithmically determining the fixed points is provided.