Dimension polynomials of extensions of difference fields
type: Item
instance of: scholarly article
title: Dimension polynomials of extensions of difference fields
zbMATH Open document ID: 0543.12017
author: I. N. Balaba
published in: Moscow University Mathematics Bulletin
publication date: 1984-01-01T00:00:00Z
review text: Let \(F\) be a difference field with the basic set of translations \(\Delta =\{\alpha_1,\ldots,\alpha_m\}\), \(\text{char}\, F=0\), \(\Gamma =\{\gamma =\alpha_1^{k_1}\cdots\alpha_m^{k_m} \mid k_1,\ldots,k_m\in\mathbb Z\}\) and \(\Gamma(s)=\{\gamma \in \Gamma \mid \text{ord}\,\gamma =\sum^{m}_{i=1}| k_i| \leq s\}\). The main result is the following: for any elements \(\eta_1,\ldots,\eta_n\) of an inversive difference field extension of \(F\) there exists an integral polynomial \(\psi\) such that for any sufficiently large integer \(s\) the transcendence degree of \(F((\gamma \eta_j),\;\gamma \in \Gamma(s),\;1\leq j\leq n)\) over \(F\) is equal to \(\psi(s)\). The polynomial \(\psi\) is called the difference dimension polynomial of the extension \(F(\eta)\) over \(F\). Some properties of dimension polynomials are described.
Mathematics Subject Classification ID: 12H10
Mathematics Subject Classification ID: 12F20
zbMATH DE Number: 3863323
zbMATH Keywords: difference field
zbMATH Keywords: inversive difference field extension
zbMATH Keywords: trancendence degree
zbMATH Keywords: difference dimension polynomial
reviewed by: E. V. Pankrat'ev
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3863323