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.

翻译：我们调查了计算一组有限生成的矩阵的Zariski封口的复杂性。Zariski封口以前被Derksen、Jeandel和Koiran证明是可比较的,但是其算法的终止理由似乎没有产生任何复杂性。在本文中,我们采取了不同的做法,并获得了定义关闭的多面体的界限。我们的封口表明,关闭可以在基本时间内计算。我们还获得了线性代数组链的长度的上限,所有组都是在固定数字字段中生成的。