On Radical Inversion and Characterization of Dickson Polynomials as Chebyshev-Type Maps

We extend a radical inversion framework from Chebyshev polynomials to the broader class of Dickson polynomials. For each integer n >= 2 and parameter α ≠ 0, we show that the equation Dₙ (y, α) =C admits an explicit radical parametrization of the form y = (A) ^ (1/n) + α/ (A) ^ (1/n), where A = C/2 + sqrt ( (C/2) ² - αⁿ). This formulation reveals that the classical Chebyshev inversion arises as a special case (α=1) of a more general algebraic structure. We further investigate structural constraints under which such parametrizations occur, leading to a characterization of polynomial maps that reduce to Dickson polynomials up to affine conjugation. Connections with complex dynamics and Kummer-type extensions are also discussed. This is a preprint that has not yet undergone peer review.