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 → tⁿ to a polynomial Pₙ satisfying Pₙ (φ_α (t) ) = φ_α (tⁿ) 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 uⁿ = A, we prove a rigidity lemma showing that any such polynomial is affinely conjugate to a Dickson polynomial Dₙ (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.
Waleed mohamed khalaf Moqadem (Tue,) studied this question.