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Polynomial mappings on Abelian groups. (English) Zbl 1073.39018

Author’s abstract: Let \(G\) and \(G'\) be Abelian groups. There are several conditions describing the property that \(f:G\to G'\) is a polynomial map of degree less than \(n\). The author investigates the relations between these conditions including the following four: (i) \(f\) is the sum of monomials of degree less than \(n\), (ii) \(\Delta_{h_1}\dots\Delta_{h_n}f\equiv 0\) for every \(h_1,\dots,h_n\in G\), (iii) \(\Delta_h^nf\equiv 0\) for every \(h\in G\), and (iv) there are functions \(f_1,\dots,f_n:G\to G'\) and integers \(a_i, b_i\) such that \(b_i\neq 0\;(i=1,\dots,n)\), and \(f(x)=\sum_{i=1}^nf_i(a_ix+b_iy)\) for every \(x,y\in G\).
It is known that (i) \(\Rightarrow\) (ii) \(\Rightarrow\) (iii) \(\Rightarrow\) (iv) holds for every \(f\). The author attempts to find the mildest conditions under which the reverse implications hold. He proves that (iii) implies (ii) supposing that, for every prime \(p\leq n\), either \(G'\) does not have elements of order \(p\), or \(| G/pG| \leq p\), and if \(G\) is divisible by every prime \(p<n\) then (ii) implies (i). It is shown also that if \(G\) is divisible, then all these conditions are equivalent to each other.

MSC:

39B52 Functional equations for functions with more general domains and/or ranges
39A12 Discrete version of topics in analysis
39A70 Difference operators
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