14.22 (1999)

Prove that any irreducible system of equations $S(x_1, \dots, x_n) = 1$ with coefficients in a torsion-free linear group $G$ is equivalent over $G$ to a finite system $T(x_1, \dots, x_n) = 1$ satisfying an analogue of Hilbert’s Nullstellensatz, i. e. $\operatorname{Rad}_G(T) = \sqrt{T}$. This is true if $G$ is a free group (O. Kharlampovich, A. Myasnikov, J. Algebra, 200 (1998), 472–570).

Here both $S$ and $T$ are regarded as subsets of $G[X] = G * F(X)$, a free product of $G$ and a free group on $X = \{x_1, \dots, x_n\}$. By definition, $\operatorname{Rad}_G(T) = \{w(x_1, \dots, x_n) \in G[X] \mid w(g_1, \dots, g_n) = 1$ for any solution $g_1, \dots, g_n \in G$ of the system $T(X) = 1\}$, and $\sqrt{T}$ is the minimal normal isolated subgroup of $G[X]$ containing $T$.