Alexey Miasnikov

The Diophantine problem in Chevalley groups

In this paper we study the Diophantine problem in Chevalley groups $G_π(Φ,R)$, where $Φ$ is an indecomposable 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 principle to show that any one-parametric subgroup $X_α$, $α\in Φ$, 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.