Separately polynomial functions
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AbstractIt is known that if $$f:{{\mathbb R}}^2\rightarrow {\mathbb R}$$ f : R 2 → R is a polynomial in each variable, then f is a polynomial. We present generalizations of this fact, when $${{\mathbb R}}^2$$ R 2 is replaced by $$G\times H$$ 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 one of them is metrizable. We present several examples showing that the results are not far from being optimal.
1970 ◽
Vol 22
(2)
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pp. 148-154
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2020 ◽
Vol 18
(04)
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pp. 2050019
1994 ◽
Vol 126
(1)
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pp. 1-6
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1993 ◽
Vol 53
(2)
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pp. 131-151
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1966 ◽
Vol 62
(3)
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pp. 399-420
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