Polynomial mappings on Abelian groups

Summary.

Let G and G ′ be Abelian groups. There are several conditions describing the property that f : G → G ′ is a polynomial map of degree less than n. We investigate the relations between these conditions including the following four: (i) f is the sum of monomials of degree less than n, (ii) $\Delta _{h_1 } \ldots \Delta _{h_n } f \equiv 0$ for every $h_1 , \ldots ,h_n \in G,$ (iii) $\Delta _h^n f \equiv 0$ for every $h \in G,$ and (iv) there are functions $f_1 , \ldots ,f_n :G \to G'$ and integers a _i , b _i such that $b_i \ne 0\;(i = 1, \ldots ,n),$ and $f(x) = \sum\nolimits_{i = 1}^n {f_i (a_i x + b_i y)}$ for every $x,y \in G.$

It is known that ${\text{(i)}} \Rightarrow {\text{(ii)}} \Rightarrow {\text{(iii)}} \Rightarrow {\text{(iv)}}$ holds for every f. We attempt to find the mildest conditions under which the reverse implications hold. We prove, for example, that (iii) implies (ii) supposing that, for every prime $p \leq n,$ either G′ does not have elements of order p, or $|G/pG| \leq p.$ We also show that if G is divisible by every prime $p \leq n$ then (ii) implies (i). As a corollary we find that if G is divisible, then all these conditions are equivalent to each other.