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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.
Young Joon Ley (Tue,) studied this question.