On a class of \(m\)-to-1 functions (Q6546423)

We say that a function \(f:U\to V\) is \(m\)-to-\(1\) if each element of \(f(U)\) has exactly \(m\) \(f\)-preimages.\N\NThe paper under review concerns Theorem 4.7 in [\textit{D. Bartoli} et al., Discrete Appl. Math. 309, 194--201 (2022; Zbl 1504.11130)]. The authors of the aforementioned article did not prove Theorem 4.7 in their paper, but proved a special case of it.\N\NThe authors in the paper under review give a simple proof of Theorem 4.7, and also refine and generalize the result in several ways. The following is their main result, which subsumes Theorem 4.7.\N\N \textbf{Theorem} Let \(q\) be a prime power, and let \(G\) be a finite subgroup of \(\mathrm{PGL}_2(\overline{\mathbb{F}_q})\). Suppose there exist distinct \(a,b\in \mathbb{F}_q\) which are both fixed by \(\tau_g(X)\) for every \(g\in G\). Write \(f(X):=\sum_{g\in G}\,\tau_g(X)\), and let \(m\) be the number of elements \(g\in G\) for which \(\tau_g(X)\) is in \(\mathbb{F}_q(X)\). Write \(n:=|G|\), and let \(S\) be the set of poles of \(F(X)\) in \(\mathbb{F}_q\). Then the following hold:\N\N\begin{itemize}\N\item[1.] \(\gcd(n,q)=1\);\N\item[2.] \(\gcd(n,q-1)=m\);\N\item[3.] \(\deg (f)=n\);\N\item[4.] \(|S|=m-1\);\N\item[5.] \(\gamma \mapsto f(\gamma)\) defines \(m\)-to-\(1\) functions \(\mathbb{P}^1(\mathbb{F}_q)\setminus \{a,b\}\to \mathbb{P}^1(\mathbb{F}_q)\) and \(\mathbb{F}_q\setminus (S\cup \{a,b\})\to \mathbb{F}_q\);\N\item[6.] We have \N\[\N\displaystyle{f(X)=\frac{n(aX-b)}{X-1}\circ X^n\circ \frac{X-b}{X-a}}.\N\]\N\end{itemize}