Maps preserving the sum-to-difference ratio in characteristic $p$

TL;DR

This paper introduces and characterizes a new group of self-maps on fields that preserve a specific ratio related to sums and differences, with explicit computations for fields algebraic over finite fields.

Contribution

It defines a novel group of maps satisfying a functional equation and computes this group for fields algebraic over finite fields, distinguishing certain finite fields.

Findings

Computed the group for all fields algebraic over $ ext{F}_p$

Distinguished $ ext{F}_5$ among finite fields and subfields of algebraic closures

Provided explicit descriptions of solutions to the functional equation

Abstract

Given a field $\mathbb{F}$, we introduce a novel group $SD(\mathbb{F})$ of its self-maps: the solutions $f \colon \mathbb{F} \twoheadrightarrow \mathbb{F}$ to the functional equation $f \left( (x+y)/(x-y) \right) = (f(x) + f(y))/(f(x) - f(y))$ for all $ x \neq y$ in $\mathbb{F}$. We compute this group for all fields algebraic over $\mathbb{F}_p$. In particular, this group distinguishes $\mathbb{F}_5$ among all finite fields $\mathbb{F}_q$, and in fact among all subfields of $\overline{\mathbb{F}_q}$.