\(E\)-algebraic functions over fields of positive characteristic -- an analogue of differentially algebraic functions
type: Item
instance of: scholarly article
title: \(E\)-algebraic functions over fields of positive characteristic -- an analogue of differentially algebraic functions
zbMATH Open document ID: 0942.12003
author: H. Sharif
published in: Journal of Algebra
publication date: 2000-08-27T00:00:00Z
review text: Let \(L\) be a differential field of characteristic zero. A function (or a power series) \(f\) is called differentially algebraic over \(L\) if it satisfies a differential equation of the form \(P(x,y,y',\dots,y^{(n)})=0\), where \(P\) is a nontrivial polynomial. This notion is not so significant over fields of characteristic \(p>0\), because \(f^{(p)}\equiv 0\). An analogue of the concept of a differentially algebraic power series is defined in the paper for a formal power series over a perfect field \(K\) of positive characteristic. It is shown that these series together with ordinary addition and multiplication of series form a field \(\Gamma_K\). The properties of this field are studied. It is shown that the field \(\Gamma_K\) is not closed under the Hadamard product operation (termwise multiplication of series).
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 11J99
Mathematics Subject Classification ID: 12H99
zbMATH DE Number: 1213797
zbMATH Keywords: differentially algebraic power series
zbMATH Keywords: differential fields of positive characteristic
reviewed by: E. V. Pankrat'ev
MaRDI profile type: Publication
full work available at URL: jabr.1998.7440
OpenAlex ID: W2092861793
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DOI: 10.1006/JABR.1998.7440
description: scientific article; zbMATH DE number 1213797