Algorithm to Compute Orbit Zariski Closure in Affine Plane

TL;DR

This paper presents an algorithm for computing the Zariski closure of orbits under algebraic group actions on the affine plane, involving group containment and invariant subvariety calculations.

Contribution

It introduces a novel algorithm that determines orbit Zariski closures by combining group containment checks and invariant subvariety computations.

Findings

Algorithm effectively computes orbit Zariski closures.

Decides group containment within algebraic groups.

Calculates invariant subvarieties for automorphisms.

Abstract

The article demonstrates the procedure how to compute the Zariski closure of an orbit by an algebraic action of finitely generated group on the affine plane. First half of the algorithm is about deciding whether given finitely generated group is contained in an algebraic group. For the next half, we compute the totality of the invariant subvarieties for a single triangular automorphism. Then, the computation for the individual generators is applied to compute the orbit Zariski closure for a finitely generated group.