Algebraic Geometry over Free Groups: Lifting Solutions into Generic Points

In this paper we prove Implicit Function Theorems (IFT) for algebraic varieties defined by regular quadratic equations and, more generally, regular NTQ systems over free groups. In the model theoretic language these results state the existence of very simple Skolem functions for particular $\mathrm{\forall }\mathrm{\exists }$-formulas over free groups. We construct these functions effectively. In non-effective form IFT first appeared in \cite{Imp}. From algebraic geometry view-point IFT can be described as lifting solutions of equations into generic points of algebraic varieties. Moreover, we show that the converse is also true, i.e., IFT holds only for algebraic varieties defined by regular NTQ systems. This implies that if a finitely generated group $H$ is $\mathrm{\forall }\mathrm{\exists }$-equivalent to a free non-abelian group then $H$ is isomorphic to the coordinate group of a regular NTQ system.