Maps preserving the sum-to-difference ratio

TL;DR

This paper characterizes all functions over various fields that satisfy a specific functional equation relating sums and differences, combining algebra, analysis, and topology in the proofs.

Contribution

It provides a complete solution to the functional equation over multiple fields, including real, rational, algebraic, and complex numbers, with continuity assumptions.

Findings

Solutions characterized for $\, ext{Q}$, $ ext{R}$, and subfields.

Results include continuous solutions over the reals.

Method combines algebra, analysis, and topology.

Abstract

For a field $\mathbb{F}$, what are all functions $f \colon \mathbb{F} \rightarrow \mathbb{F}$ that satisfy 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 solve this problem for the fields $\mathbb{Q}, \mathbb{R}$, and a class of its subfields that includes the real constructible numbers, the real algebraic numbers, and all quadratic number fields. We also solve it over the complex numbers and on any subfield of $\mathbb{R}$, if $f$ is continuous over the reals. The proofs involve a mix of algebra in all fields, analysis over the real line, and some topology in the complex plane.