Complete sets in differentially closed fields
type: Item
instance of: scholarly article
title: Complete sets in differentially closed fields
zbMATH Open document ID: 1037.12009
author: Wai Yan Pong
published in: Journal of Algebra
publication date: 2001-10-24T00:00:00Z
review text: Using basic model theory and differential algebraic geometry, the author studies complete sets in the frame of differential algebraic geometry. Let \((F, \delta)\) be a fixed differentialy closed field of characteristic \(0\), with the field of constants \(C\). All \(\delta\)-varieties are identified with the sets of their \(F\)-points. A \(\delta\)-closed set \(X\) in the projective space \(\mathbb{P}^n\) over \(F\) is \(\delta\)-complete if the second projection \(p_2 : X \times Y \to Y\) is a \(\delta\)-closed map for every quasi-projective \(\delta\)-variety \(Y\). By contrast with the situation in algebraic geometry, the author shows that every \(\delta\)-complete set is affine and is definably isomorphic to a \(\delta\)-complete subset of the affine line. Next he obtains a valuative criterion for \(\delta\)-completeness, and, using this criterion, he obtains Kolchin's result on \(\delta\)-completeness of \(\mathbb{P}^n(C)\) and finds a new family of \(\delta\)-complete sets.
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 12L12
Mathematics Subject Classification ID: 14A99
zbMATH DE Number: 1428250
zbMATH Keywords: differentially closed field
zbMATH Keywords: \(\delta\)-complete
zbMATH Keywords: positive quantifier free formula
reviewed by: Şerban A. Basarab
MaRDI profile type: Publication
full work available at URL: jabr.1999.8077
OpenAlex ID: W1971016543
cites work: Q5563439
cites work: Complete Models of Differential Fields
cites work: Extending Differential Specializations
cites work: Q5691061
cites work: The classification of the semisimple differential algebraic groups and the linear semisimple differential algebraic Lie algebras
cites work: Q4143433
cites work: Q5680222
cites work: Q4043179
cites work: Q3801533
cites work: Q4872381
cites work: Extensions of Differential Places
cites work: Differential Specializations in Integral and Algebraic Extensions
cites work: Some applications of ordinal dimensions to the theory of differentially closed fields
cites work: Q3829645
cites work: Q3324077
DOI: 10.1006/JABR.1999.8077
Recommended article: Q4212158
Recommended article: On differentially closed ordered fields
Recommended article: On model complete differential fields
Recommended article: The number of countable differentially closed fields
Recommended article: Maximally complete fields
Recommended article: Quite complete real closed fields
Recommended article: Q3136938
Recommended article: Q3724950
Recommended article: Q4335126
Recommended article: Q3764243
description: scientific article; zbMATH DE number 1428250