Polynomial functions on Abelian groups play a basic role in the theory of functional equations and spectral analysis. In this paper we investigate the ring-structure of polynomial functions on topological Abelian groups. We show that polynomial functions form a Noetherian ring if and only if the linear space of continuous homomorphisms of the group into the additive group of complex numbers is finite dimensional. In the case of discrete Abelian groups this is equivalent to the presence of spectral synthesis.
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The research was supported by the Hungarian National Foundation for Scientific Research (OTKA), Grant No. NK-81402.
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Székelyhidi, L. Noetherian rings of polynomial functions on Abelian groups. Aequat. Math. 84, 41–50 (2012). https://doi.org/10.1007/s00010-011-0107-9