An invariant for difference field extensions
type: Item
instance of: scholarly article
title: An invariant for difference field extensions
zbMATH Open document ID: 1250.12005
author: Zoé Chatzidakis
author: Ehud Hrushovski
published in: Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI
publication date: 2012-08-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/0902.0844
reviewed by: Alexander B. Levin
Mathematics Subject Classification ID: 12H10
Mathematics Subject Classification ID: 03C60
zbMATH DE Number: 6062790
zbMATH Keywords: Difference field
zbMATH Keywords: distant degree
zbMATH Keywords: limit degree
zbMATH Keywords: difference subgroup
MaRDI profile type: Publication
OpenAlex ID: W2963410724
arXiv ID: 0902.0844
cites work: Q5334359
cites work: The problem of finite axiomatizability for strongly minimal graph theories
cites work: Structure Theory of Totally Disconnected Locally Compact Groups via Graphs and Permutations
cites work: Q3130253
cites work: The structure of totally disconnected, locally compact groups
cites work: Further properties of the scale function on a totally disconnected group
DOI: 10.5802/AFST.1334
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review text: The paper under review introduces a new invariant of an ordinary difference field extension, called the \textit{distant degree} of the extension, and studies properties of this invariant. Let \(K\) be an ordinary difference field with a distinguished endomorphism \(\sigma\). If \(a\) is a finite tuple in some difference field extension of \(K\) such that \(\sigma(a)\) lies in the algebraic closure of \(K\), then the distant degree and inverse distant degree of \(a\) over \(K\) are defined as \(dd(a/K) = \lim_{k\rightarrow\infty}[K(a, \sigma^{k}(a)):K(a)]^{1/k}\) and \(idd(a/K) = \lim_{k\rightarrow\infty}[K^{\text{inv}}(a, \sigma^{-k}(a)):K^{\text{inv}}(a)]^{1/k}\), respectively. It is shown that if \(a\) and \(b\) are tuples over \(K\) (the coordinates of \(a\) and \(b\) lie in some large difference overfield \(\mathcal{U}\) of \(K\)) such that the algebraic closure \(K(a)_{\sigma}^{\text{alg}}\) of the difference field \(K(a)_{\sigma}\) generated by \(a\) over \(K\) coincides with \(K(b)_{\sigma}^{alg}\), then \(dd(a/K) = dd(b/K)\). Also, for any tuple \(a\) over \(K\), \(dd(a/K) = dd(a/K^{inv})\) and \(dd(a/K)ild(a/K) = idd(a/K)ld(a/K)\) where \(ld(a/K)\) and \(ild(a/K)\) denote the limit degree and inverse limit degree of \(K(a)_{\sigma}\) over \(K\). (The limit degree and inverse limit degree are characteristics of a difference field extension introduced in [\textit{R. M. Cohn}, ``An invariant of difference field extensions'', Proc. Am. Math. Soc. 7, 656--661 (1956; Zbl 0070.27001)], see also Chapter 5 of [\textit{R. M. Cohn}, Difference algebra. New York etc.: Interscience Publishers (1965; Zbl 0127.26402)] and Chapter 4 of [\textit{A. Levin}, Difference algebra. Algebra and Applications 8. New York, NY: Springer (2008; Zbl 1209.12003)]). Furthermore, the authors prove that there exists a constant \(m = m(a, K)\) such that \(dd(a/K) = \frac{ld(a/K)}{m}\) (assuming \(\sigma(a)\in K(a)^{\text{alg}}\) and \(K(a, \sigma(a)):K(a) = ld(a/K)\)), and establish some oth…
description: scientific article; zbMATH DE number 6062790