Algorithm to Compute Orbit Zariski Closure in Affine Plane

The paper proves decidability and gives a complete, case-by-case algorithm that handles unbounded subgroups via Blanc–Stampfli and Whang, and bounded subgroups via an effective conjugacy test into Aff or J followed by a full classification of invariant subvarieties, including the crucial “projective quotient” case for triangular automorphisms (π: A^2 ↠ P^1). The candidate solution omits this projective-quotient branch. For example, for g(x,y)=(2x,4y) (a,b not roots of unity but with a^2=b), the paper’s algorithm correctly finds π(x,y)=[x^2:y] and computes the orbit-closure as a fiber such as y=x^2, whereas the model’s procedure would wrongly output A^2 after failing to find a polynomial graph x=H(y).