Separately polynomial functions

TL;DR

This paper generalizes the classical polynomial characterization in two variables to functions on products of topological Abelian groups, identifying conditions under which such functions are generalized polynomials.

Contribution

It extends polynomial function results from Euclidean spaces to topological Abelian groups, providing new conditions for functions to be generalized polynomials.

Findings

The polynomial property holds when G is a connected Baire space and H has a dense subgroup of finite rank.

For continuous functions, the property holds if G and H are connected Baire spaces.

Continuity can be omitted if G and H are locally compact or complete metric spaces.

Abstract

It is known that if $f\colon {\mathbb R}^2 \to {\mathbb R}$ is a polynomial in each variable, then $f$ is a polynomial. We present generalizations of this fact, when ${\mathbb R}^2$ is replaced by $G\times H$, 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 complete metric spaces. We present several examples showing that the results are not far from being optimal.