标题： 保持和差比的映射 显示英文标题

标题： Maps preserving the sum-to-difference ratio

摘要： 对于域$\mathbb{F}$，所有满足泛函方程$f \left( (x+y)/(x-y) \right) = (f(x) + f(y))/(f(x) - f(y))$的函数$f \colon \mathbb{F} \rightarrow \mathbb{F}$是什么，其中所有$ x \neq y$在$\mathbb{F}$中？ 我们解决了域$\mathbb{Q}, \mathbb{R}$以及包含实构造型数、实代数数和所有二次数域的其子域类的问题。 我们也在复数上解决它，并在$\mathbb{R}$的任何子域上解决，如果$f$在实数上是连续的。 证明涉及所有域中的代数、实数轴上的分析以及复平面上的一些拓扑学。 显示英文摘要

摘要： 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.