Expansions of algebraically closed fields in o-minimal structures
type: Item
instance of: scholarly article
title: Expansions of algebraically closed fields in o-minimal structures
zbMATH Open document ID: 1010.03027
DOI: 10.1007/PL00001405
author: S. Starchenko
author: Ya'acov Peterzil
published in: Selecta Mathematica. New Series
publication date: 2002-08-25T00:00:00Z
review text: An algebraically closed field \(K\) can be conceived as \(R( \sqrt{-1})\), with real closed \(R\). And, \(R\) is orderable by the Artin-Schreier theorem, and so o-minimality is relevant to \(R\), \(K\), and their expansions. In a long first part of this paper, the authors develop a differential part of complex analysis on \(K\). Classical tools, like convergence and integrals are not available, and they resort to topological tools like winding numbers. (The main reference is \textit{G. T. Whyburn}'s book: Topological analysis. Revised ed. Princeton, N. J.: Princeton Univ. Press (1964; Zbl 0186.55901), the authors state.) Their development is in terms of definable sets and functions, and includes (analogues of) Laurent expansions, residues, the maximum principle, and Liouville's and Rouché's theorems. These works are used to obtain a model-theoretic result, as follows. Let \({\mathcal R}\) be an o-minimal expansion of \(R\), and let \({\mathcal K}\) be the expansion of \(K\) defined in \({\mathcal R}\). If \({\mathcal K}\) defines more sets than \(\langle K,+,\cdot\rangle\) does, then \({\mathcal K}\) defines \(R\), too. From this follows a classical result of Chow: An analytic set in \(\mathbb{P}_n (\mathbb{C})\) is algebraic. (For the usual proof, refer, e.g., to \textit{P. Griffiths} and \textit{J. Harris}, Principles of algebraic geometry, New York etc.: Wiley (1978; Zbl 0408.14001)).
Mathematics Subject Classification ID: 03C64
Mathematics Subject Classification ID: 12L12
Mathematics Subject Classification ID: 14A10
Mathematics Subject Classification ID: 30C35
Mathematics Subject Classification ID: 12J15
Mathematics Subject Classification ID: 03H05
zbMATH DE Number: 1787782
zbMATH Keywords: o-minimality
zbMATH Keywords: winding number
zbMATH Keywords: Chow's theorem
zbMATH Keywords: real closed field
zbMATH Keywords: algebraically closed field
zbMATH Keywords: complex analysis
zbMATH Keywords: o-minimal expansion
zbMATH Keywords: analytic set
MaRDI profile type: Publication
Recommended article: Q3710789
Recommended article: Q3994501
Recommended article: Conferenze di geometria algebrica.
Recommended article: Q4001976
Recommended article: Q3995751
Recommended article: Géométrie sur les surfaces et variétés algébriques.
Recommended article: Q4692956
Recommended article: Q3322387
Recommended article: Topology of nonarchimedean analytic spaces and relations to complex algebraic geometry
description: scientific article; zbMATH DE number 1787782