Polynomial mappings of finitely generated \({\mathbb{Z}}\)-modules
type: Item
instance of: scholarly article
title: Polynomial mappings of finitely generated \({\mathbb{Z}}\)-modules
zbMATH Open document ID: 0619.08003
DOI: 10.1016/0022-314X(87)90087-4
author: Jordan Hrycaj
published in: Journal of Number Theory
publication date: 1987-01-01T00:00:00Z
review text: 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 \(\Phi^ 0_ k(G;H)=\{f\in \Phi_ k(G;H)|\) \(f(0)=0\}\).
Mathematics Subject Classification ID: 08A40
Mathematics Subject Classification ID: 11B39
Mathematics Subject Classification ID: 20K99
Mathematics Subject Classification ID: 13B25
zbMATH DE Number: 4004246
zbMATH Keywords: polynomial mapping
MaRDI profile type: Publication
cites work: On cubic polynomials. II: Multiple exponential sums
description: scientific article; zbMATH DE number 4004246