Maps preserving the sum-to-difference ratio

TL;DR

This work classifies maps on fields that preserve the sum-to-difference ratio under the transform $f\left(\frac{x+y}{x-y}\right) = \frac{f(x)+f(y)}{f(x)-f(y)}$. Using a mix of algebraic and analytic techniques, the authors show that SD maps are heavily constrained: over $\mathbb{Q}$, $\mathbb{R}$, and Euclidean subfields the only SD maps are the identity; over quadratic fields the maps are restricted to the identity or complex conjugation; continuous or $\mathbb{R}$-preserving SD maps on subfields of $\mathbb{C}$ containing $\mathbb{Q}(i)$ likewise reduce to identity or conjugation. They also present SD maps that are not automorphisms, highlighting that the equation does not force additivity or surjectivity in general. The results bridge algebra, analysis, and topology and raise questions about SD maps in broader field contexts, including finite fields.

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.

Theorems & Definitions (39)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Proposition 2.1
• proof
• Remark 2.2
• Theorem 3.1
• proof