Diophantine geometry over groups. IV: An iterative procedure for validation of a sentence.
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.
Makanin-Razborov diagrams: Diophantine sets
scientific article; zbMATH DE number 2141369