The Diophantine problem in the classical matrix groups

In this paper we study the Diophantine problem in the classical matrix groups GLn(R), SLn(R), Tn(R) and UTn(R), n ≥ 3, over an associative ring R with identity. We show that if Gn(R) is one of these groups, then the Diophantine problem in Gn(R) is polynomial-time equivalent (more precisely, Karp equivalent) to the Diophantine problem in R. When Gn(R) = SLn(R) we assume that R is commutative. Similar results hold for PGLn(R) and PSLn(R) provided R has no zero divisors (for PGLn(R) the ring R is not assumed to be commutative).