Perfect pseudo-algebraically closed fields are algebraically bounded.
type: Item
instance of: scholarly article
title: Perfect pseudo-algebraically closed fields are algebraically bounded.
zbMATH Open document ID: 1046.12002
author: Zoé Chatzidakis
author: Ehud Hrushovski
published in: Journal of Algebra
publication date: 2004-02-03T00:00:00Z
review text: Algebraic boundedness for theories \(T\) of fields is a concept that was introduced by L. Van den Dries, regarding, for instance, algebraically closed fields, real closed fields and \(p\)-adic fields, and ensuring a well-behaved notion of dimension on definable sets \(S\) of models of \(T\) (essentially coinciding with the algebraic dimension of the Zariski closure of \(S\)). In fact, \(T\) is said to be algebraically bounded if and only if for every formula \(\varphi(\overline{x}, \, y)\) there are finitely many polynomials \(f_j (\overline{x}, \, y)\) with coefficients from \({\mathbb Z}\) such that, for any model \(K\) of \(T\) and \(t\)-tuple \(\overline{a}\) in \(K\), there is some \(j\) for which \(f_j(\overline{a}, \, y)\) is not identically \(0\) in \(K\) and \(\varphi(\overline{a}, \, K)\) is just contained in the solution set of \(f_j(\overline{a}, \, y) = 0\) in \(K\). So, algebraic boundedness is a strong uniformity notion on finite definable sets. The authors consider here pseudo-algebraically closed fields (i.e., fields \(K\) where every absolutely irreducible variety over \(K\) has a \(K\)-rational point), so a quite intriguing framework in Model Theory, and show that every theory of perfect pseudo-algebraically closed fields is algebraically bounded. This extends a previous result of Jarden on perfect Frobenius fields.
Mathematics Subject Classification ID: 12E30
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 12L99
zbMATH DE Number: 2037104
zbMATH Keywords: pseudo-algebraically closed field
zbMATH Keywords: algebraically bounded theory
reviewed by: Carlo Toffalori
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2002.11.006
OpenAlex ID: W2067170342
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DOI: 10.1016/J.JALGEBRA.2002.11.006
description: scientific article; zbMATH DE number 2037104