On the Computation of the Zariski Closure of Finitely Generated Groups of Matrices

TL;DR

This paper provides a new complexity bound for computing the Zariski closure of finitely generated matrix groups, demonstrating that the process can be performed in elementary time with explicit polynomial degree bounds.

Contribution

It introduces a novel approach that establishes an explicit bound on the degree of defining polynomials, ensuring elementary time computability of the Zariski closure.

Findings

Bound on the degree of defining polynomials for the Zariski closure

Elementary time complexity for computing the closure

Upper bounds on chain lengths of linear algebraic groups

Abstract

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.