  We investigate the complexity of computing the Zariski closure of a finitely
generated group of matrices. The Zariski closure was previously shown to be
computable by Derksen, Jeandel, and Koiran, but the termination argument for
their algorithm appears not to yield any complexity bound. In this paper we
follow a different approach and obtain a bound on the degree of the polynomials
that define the closure. Our bound shows that the closure can be computed in
elementary time. We also obtain upper bounds on the length of chains of linear
algebraic groups, where all the groups are generated over a fixed number field.
