Polynomial mappings of finitely generated \({\mathbb{Z}}\)-modules (Q1089369)

Let \(f: G\to H\) be a map between two abelian groups G and H, and let \(k\in {\mathbb{N}}_ 0\). Then f is a polynomial mapping of degree \(\leq k\) if \((\prod^{k+1}_{\nu =1}\Delta_{a_ v})f=0\) for all \(a_ 1,...,a_{k+1}\in G\), where \((\Delta_ af)(x)=f(x+a)-f(x)\) (\(\forall x\in G)\). The least such \(k\in {\mathbb{N}}\) is called the degree of f. Let \(\Phi_ k(G;H)\) be the set of polynomial mappings of degree \(\leq k\); then \(\Phi_ k(G;H)\) is a module. The subject of this paper is the study of this module. If \(\Delta_ af=0\) (\(\forall a\in G)\), then f is constant, \(\Phi_ 0(G;H)\cong H\). For \(k=1\) we have \(\Phi_ 1(G,H)\cong H\oplus Hom(G,H)\). In general \(\Phi_ k(G;H)=\Phi_ 0(G;H)\oplus \Phi^ 0_ k(G;H)\) where \(\Phi^ 0_ k(G;H)=\{f\in \Phi_ k(G;H)|\) \(f(0)=0\}\).