The Diophantine problem in Chevalley groups

In this paper we study the Diophantine problem in Chevalley groups G_π(Φ,R), where Φ is a reduced irreducible root system of rank >1, R is an arbitrary commutative ring with 1. We establish a variant of double centralizer theorem for elementary unipotents x_α(1). This theorem is valid for arbitrary commutative rings with 1. The result is principal to show that any one-parametric subgroup X_α, α∈Φ, is Diophantine in G. Then we prove that the Diophantine problem in G_π(Φ,R) is polynomial time equivalent (more precisely, Karp equivalent) to the Diophantine problem in R. This fact gives rise to a number of model-theoretic corollaries for specific types of rings.