Separately polynomial functions

Abstract It is known that if $$f:{{\mathbb R}}^2\rightarrow {\mathbb R}$$ f : R 2 → a polynomial in each variable, then f polynomial. We present generalizations of this fact, when $${{\mathbb R}}^2$$ replaced by $$G\times H$$ G × H , where G and H are topological Abelian groups. show, e.g., the conclusion holds (with generalized polynomials place polynomials) connected Baire space has dense subgroup finite rank or, for continuous functions, spaces. The condition continuity can be omitted locally compact or one them metrizable. several examples showing results not far from being optimal.