Separately polynomial functions

Abstract

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