Diophantine geometry over groups. IV: An iterative procedure for validation of a sentence.
type: Item
instance of: scholarly article
title: Diophantine geometry over groups. IV: An iterative procedure for validation of a sentence.
zbMATH Open document ID: 1088.20017
DOI: 10.1007/BF02803494
published in: Israel Journal of Mathematics
publication date: 2005-03-07T00:00:00Z
review text: This long and carefully written paper is the fourth in the author's impressive series on the structure of sets of solutions to systems of equations in a free group [for part II cf. Isr. J. Math. 134, 173-254 (2003; Zbl 1028.20028)]. The structural results obtained in the first papers in the sequence are applied to analyze \(\forall\exists\) sentences. An iterative procedure is associated with an \(\forall\exists\) sentence to produce a sequence of varieties and formal solutions defined over them. The author proves that the complexity of Diophantine sets associated with these varieties produced along the procedure strictly increases. It follows that the procedure terminates after finitely many steps. The outcome of the iterative procedure can be viewed as a stratification theorem generalising Merzlyakov's known result from positive sentences to general \(\forall\exists\) ones. On every stratum an \(\forall\exists\) sentence can be validated using a finite set of formal solutions.
Mathematics Subject Classification ID: 20F10
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20E10
Mathematics Subject Classification ID: 20A15
Mathematics Subject Classification ID: 11D72
Mathematics Subject Classification ID: 14A22
Mathematics Subject Classification ID: 03B25
Mathematics Subject Classification ID: 03C07
Mathematics Subject Classification ID: 03C60
zbMATH DE Number: 2141369
zbMATH Keywords: equations over groups
zbMATH Keywords: solutions to systems of equations
zbMATH Keywords: free groups
zbMATH Keywords: elementary sets
zbMATH Keywords: positive sentences
zbMATH Keywords: quantifier elimination
zbMATH Keywords: Diophantine geometry over groups
zbMATH Keywords: Makanin-Razborov diagrams: Diophantine sets
reviewed by: Vitaliĭ Roman'kov
MaRDI profile type: Publication
cites work: Subgroups of Surface Groups are Almost Geometric
cites work: Diophantine geometry over groups. I: Makanin-Razborov diagrams
cites work: Diophantine geometry over groups. II: Completions, closures and formal solutions
cites work: Diophantine geometry over groups. III: Rigid and solid solutions.
author: Zlil Sela
description: scientific article; zbMATH DE number 2141369