A statement of Weierstrass on meromorphic functions which admit an algebraic addition theorem

Abstract

A statement of Weierstrass is known for meromorphic functions which admit an algebraic addition theorem. We give its precise formulation and prove it complex analytically. In fact, we show that if $K$ is a non-degenerate algebraic function field in $n$ variables over $\mathbf{C}$ which admits an algebraic addition theorem, then any $f\in K$ is a rational function of some coordinate functions and abelian functions of other variables.