Deciding if a variety forms an algebraic group

TL;DR

This paper presents an algorithm to determine whether a given algebraic variety defined by polynomials in matrix entries forms an algebraic group under matrix multiplication.

Contribution

It introduces a method to decide if the invertible matrices in a polynomial-defined variety constitute a linear algebraic group.

Findings

Algorithm successfully identifies algebraic groups within polynomial varieties.

Decides group structure efficiently for varieties in $GL(n,K)$.

Provides a computational tool for algebraic group recognition.

Abstract

Let $n$ be a positive integer and let $f_1, \ldots, f_r$ be polynomials in $n^2$ indeterminates over an algebraically closed field $K$. We describe an algorithm to decide if the invertible matrices contained in the variety of $f_1, \ldots, f_r$ form a subgroup of $GL(n,K)$; that is, we show how to decide if the polynomials $f_1, \ldots, f_r$ define a linear algebraic group.