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 \bm{C} which admits an algebraic addition theorem, then any f ∈ K is a rational function of some coordinate functions and abelian functions of other variables.