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 describe several applications of this result, e.g., concerning quantum automata and quantum universal gates. We also obtain an upper bound on the length of a strictly increasing chain of linear algebraic groups, all of which are generated over a fixed number field.

中文翻译：

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关于有限生成矩阵群的代数闭包的计算

我们研究了计算有限生成矩阵组的 Zariski 闭包的复杂性。此前，Derksen、Jeandel 和 Koiran 已经证明 Zariski 闭包是可计算的，但他们算法的终止参数似乎没有产生任何复杂性界限。在本文中，我们采用不同的方法并获得定义闭包的多项式的次数的界限。我们的界限表明可以在基本时间内计算闭包。我们描述了这个结果的几个应用，例如，关于量子自动机和量子通用门。我们还获得了严格递增的线性代数群链的长度上限，所有这些链都是在固定数域上生成的。