Dimension polynomials of intermediate fields and Krull-type dimension of finitely generated differential field extensions
type: Item
instance of: scholarly article
title: Dimension polynomials of intermediate fields and Krull-type dimension of finitely generated differential field extensions
zbMATH Open document ID: 1230.12004
author: Alexander B. Levin
published in: Mathematics in Computer Science
publication date: 2011-06-08T00:00:00Z
review text: Let \(K\) be a (partial) differential field of characteristic \(0\) with set of derivations \(\Delta\), let \(L=K\langle\eta_1, \dots, \eta_s\rangle\) be a finitely generated differential field extension of \(K\) and let \(F\) be an intermediate differential field. Generalizing Kolchin's differential dimension polynomial, the author shows that there is a rational polynomial integral on the integers which counts the transcendence degree of \(F_r\) over \(K\), where \(F_r\) is the intersection of \(F\) with the extension of \(K\) algebraically generated by derivatives of the \(\eta_i\) up to order \(r\). As in Kolchin's case, the polynomial depends on the \(\eta_i\)'s, but not its degree, \(d\), and leading coefficient \(b_d\). Let \(m=| \Delta |\). Then \(d \leq m\); it is convenient to write the polynomial including terms up to degree \(m\), in which case the coefficient \(b_m\) is also an invariant: either \(d < m\) and \(b_m=0\) or \(b_m=d\). In any event, \(b_m\) is the differential transcendence degree of \(F\) over \(K\). The theorem is proved via a consideration of Hilbert functions of differential modules. Retain the above notation for \(L\) and \(K\). The author further considers the set \(\mathfrak U\) of pairs \((F,E)\) of intermediate differential fields with \(F \supseteq E\). The author shows that there is a unique mapping \(\mu: \mathfrak U \to \mathbb Z \cup \{\infty\}\), which takes values \(-1\) or larger, and such that if \(k\) is any natural number then \(\mu(F,E) \geq k\) if and only if \(F\) is not algebraic over \(E\) and there exists an infinite proper descending chain \(\{ F_i\}\) of intermediate differential fields with \(F_0=F\), \(F_i \supseteq E\) all \(i\), and \(\mu(F_i,F_{i+1}) \geq k-1\) all \(i\). He then defines the differential transcendence type of \(L\) over \(K\), \(\Delta\text{-tr.type}(L/K)\), to be the supremum of \(\mu\). He also defines the differential transcendence dimension of \(L\) over \(K\), \(\Delta-tr.dim(L/K)\), to…
reviewed by: Andy R. Magid
Mathematics Subject Classification ID: 12H05
zbMATH DE Number: 5905248
zbMATH Keywords: differential field
zbMATH Keywords: differential type
zbMATH Keywords: differential transcendence degree
MaRDI profile type: Publication
full work available at URL: s11786-010-0048-z
OpenAlex ID: W1965560035
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DOI: 10.1007/S11786-010-0048-Z
description: scientific article; zbMATH DE number 5905248