Two-to-one functions from Galois extensions
An \(m\)-1 function over a finite field \(\mathbb{F}_q\) is a function \(f:\mathbb{F}_q\rightarrow\mathbb{F}_q\) such that for each \(b\in\mathbb{F}_q, |f^{-1}(b)|\in\{0, m\}\), except for at most a single value \(\in\mathbb{F}_q\) for which \(|f^{-1}(b)|\notin\{0, m\}\). The main achievement of the paper is the construction of new 2-1 rational functions in odd characteristic. The construction is relatively straightforward, related to algebraic curves over finite fields and Galois groups of rational functions. It is based on rational function \(f_{a,b}(X)=N(X)/D(X)\) associated to the group \(C_{2d}=\{ g\left(\begin{smallmatrix}1&0\\ 0&\delta^i\end{smallmatrix}\right)g^{-1}\mid i\in\{0,\dots,2d-1\}\}\). The authors also extend the notion of 2-1 mapping to that of \(m\)-1 mapping and provide explicit descriptions of 3-1 rational functions.
scientific article; zbMATH DE number 7456391