Integrality and Thurston Rigidity for Bicritical PCF Polynomials

The paper proves, by an arithmetic method, that for bicritical polynomials the two critical-periodicity curves intersect transversely for all but finitely many pairs (d,k); the finite exceptional set comes from the need to find an index-divisor-free (IDF) prime, which they show exists except in finitely many cases, and they conjecture essentially one remaining exception. They implement this by parametrizing the bicritical locus via dynamical Belyi polynomials, proving p-adic integrality of PCF parameters at an IDF prime, and showing the Jacobian is nonzero modulo p, hence transversality (see their Theorem 1 and Proposition 19; outline and setup in their Sections 1–3) . Independently, the candidate solution invokes the complex-analytic Thurston–Epstein transversality framework in the marked-critical moduli, noting that flexible Lattès are excluded in the polynomial locus, to conclude full transversality with no exceptions. The paper itself acknowledges that transversality is a consequence of Thurston rigidity in P^crit_d (context in their introduction) while providing a new algebraic proof in the bicritical case with a finite-exception caveat tied to IDF primes . Hence: both are correct, but they use different methods and state different scopes; the model proves a stronger statement via analytic tools, while the paper supplies an arithmetic proof with a finite-exception qualifier.