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

2021-06-03

International Symposium on Symbolic and Algebraic Computation

AI Summary

This paper addresses the computational complexity of computing the Zariski closure of a finitely generated matrix group. We establish the first explicit degree bound on the defining ideal of the closure and prove that it is constructible in elementary time—a significant advance over prior algorithms, which guaranteed computability but provided no complexity analysis. Our approach integrates tools from algebraic geometry, effective model theory, and the theory of linear algebraic groups, leveraging polynomial ideals, invariant theory, and induction on group chains. The main contributions are: (i) a proof of elementary-time computability of the Zariski closure; (ii) a uniform upper bound on the length of nested chains of linear algebraic groups over a fixed number field; and (iii) the first quantitative characterization of the structural depth of algebraic groups, transforming abstract existence results into explicit, efficiency-guaranteed constructions.

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.

Problem

• Computing Zariski closure of matrix groups
• Establishing complexity bounds for closure computation
• Determining upper bounds for algebraic group chains

Innovation

• Computes Zariski closure of matrix groups
• Bounds degree of defining polynomials
• Establishes elementary time complexity