分别多项式函数
Separately polynomial functions
摘要：
已知如果$f\colon {\mathbb R}^2 \to {\mathbb R}$是每个变量的多项式，则$f$是多项式。当${\mathbb R}^2$被$G\times H$取代时，我们给出了这个事实的推广，其中$G$和$H$是拓扑阿贝尔群。我们显示，例如，如果$G$是连通Baire空间且$H$具有有限秩的稠密子群，或者对于连续函数，如果$G$和$H$是连通Baire空间，则结论成立（用广义多项式代替多项式）。如果$G$和$H$是局部紧的或完全度量空间，则连续性条件可以省略。我们提出了几个例子表明，结果是不远处是最佳的。
英文摘要：
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.
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