A Unified Framework for Radical Parametrizations of Polynomial Maps via Symmetric Rational Lifting

We present a unified framework for radical parametrizations of polynomial maps arising from symmetric rational liftings. Building on previous work on Chebyshev and Dickson polynomials, we isolate the structural mechanism that enables radical inversion: the existence of a rational map φ_α(t) = t + α/t that lifts the power map t → t^n to a polynomial P_n satisfying P_n(φ_α(t)) = φ_α(t^n) as an identity in the function field C(t). Within the class of polynomial maps that admit a symmetric rational lifting of the form y = u + α/u with u^n = A, we prove a rigidity lemma showing that any such polynomial is affinely conjugate to a Dickson polynomial D_n(x,α). This framework captures a large class of polynomial families where radical inversion is possible. This is a preprint that has not yet undergone peer review.