, Volume 134, Issue 1, pp 173-254

Diophanting geometry over groups II: Completions, closures and formal solutions

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Abstract

This paper is the second in a series on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined over a free group. In the second paper we generalize Merzlyakov’s theorem on the existence of a formal solution associated with a positive sentence [Me]. We first construct a formal solution to a generalAE sentence which is known to be true over some variety, and then develop tools that enable us to analyze the collection of all such formal solutions.

Partially supported by an Israel Academy of Sciences Fellowship.