2021/01/08 by Kiss, Gergely, Laczkovich, Miklós · 1 citation
#22A20 #54E45 #54E52 #FOS: Mathematics #General Topology (math.GN)
paper · doi:10.48550/arxiv.2101.03094
It is known that if f\colon \mathbb R2 → \mathbb R is a polynomial in each variable, then f is a polynomial. We present generalizations of this fact, when \mathbb R2 is replaced by G× 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.