Computing the Zariski closure of a finitely generated matrix group

TL;DR

This paper presents an algorithm to compute the Zariski closure of finitely generated matrix groups, avoiding Groebner bases, and demonstrates its implementation in the OSCAR system for practical use.

Contribution

The paper introduces a novel algorithm for computing algebraic closures of matrix groups that is efficient and applicable to complex examples, implemented in OSCAR.

Findings

Algorithm successfully computes Zariski closures of matrix groups.

Implementation in OSCAR demonstrates practical usability.

Avoids computationally intensive Groebner basis calculations.

Abstract

We describe an algorithm for determining the algebraic subgroup of GL(n,C) that is defined as the closure of the group generated by a finite number of elements of GL(n,C). The algorithm avoids the use of Groebner bases and can be used on non-trivial examples. In the last section we report on an implementation of the algorithm in the computer algebra system {\tt OSCAR}.