Hurwitz series as formal functions
type: Item
instance of: scholarly article
title: Hurwitz series as formal functions
zbMATH Open document ID: 0978.12007
DOI: 10.1016/S0022-4049(98)00099-1
author: William F. Keigher
author: F. Leon Pritchard
published in: Journal of Pure and Applied Algebra
publication date: 2001-04-03T00:00:00Z
review text: Let \(A\) be a commutative ring with identity. The ring \(HA\) of Hurwitz series over \(A\) consists of countable sequences in \(A\) of the form \((a_0,a_1,a_2,\dots)\), or more simply \((a_n)\), where \(a_n\in A\) and \(n\in\mathbb N\). The ring operations are defined by \((a_n)+(b_n)=(a_n+b_n)\) and \((a_n)(b_n)=(c_n)\) where \(c_n=\sum_{k=0}^nC_k^na_kb_{n-k}\). The derivation \(\partial_A: HA\to HA\) is given by \(\partial_A((a_0,a_1,a_2,\dots))=(a_1,a_2,a_3,\dots)\). The authors continue the study, begun in [Commun. Algebra 25, 1845-1859 (1997; Zbl 0884.13013)], of properties of the ring \(HA\). In particular, they show that with its ``natural'' topology, \(HA\) is a complete metric space, and all of the natural mappings are continuous. There is a composition which satisfies the usual properties, i.e., it is continuous, associative, has a two-sided identity, is additive and multiplicative in the left (outer) factor, and satisfies the chain rule. It is also shown that \(HA\) provides formal solutions to homogeneous linear ordinary differential equations, using Picard's method of successive approximations. It is noteworthy that these results hold independently of the characteristic of the ring \(A\).
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 13N05
Mathematics Subject Classification ID: 13A35
Mathematics Subject Classification ID: 13J05
Mathematics Subject Classification ID: 13J10
Mathematics Subject Classification ID: 13F25
zbMATH DE Number: 1420405
zbMATH Keywords: differential ring
zbMATH Keywords: metric space
zbMATH Keywords: Hurwitz series
zbMATH Keywords: formal functions
zbMATH Keywords: formal solution
reviewed by: E. V. Pankrat'ev
MaRDI profile type: Publication
full work available at URL: s0022-4049(98)00099-1
OpenAlex ID: W1988386783
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description: scientific article; zbMATH DE number 1420405