On algebraic dependencies between Poincaré functions

The paper proves Theorem 1.1 using invariant-curve classification (via Medvedev–Scanlon–Pakovich machinery) and a reduction from non‑special to non‑generalized Lattès maps; it deduces genus(C)=0 and characterizes when an algebraic relation f(P_{A1}(z^{d1}),P_{A2}(z^{d2}))=0 can occur. The model’s solution reaches the same conclusions but by a different route: it uses uniformization to rule out genus 1 (since that would force periodic/doubly periodic Poincaré functions and hence special maps), and uses Koenigs–Poincaré linearization plus uniqueness to derive the semiconjugacy characterization. Minor stylistic differences (e.g., asserting k=1 in one direction) do not affect correctness; overall, both arguments are valid and compatible with the paper’s statements.