Radical Inversion of Chebyshev-Type Polynomial Maps and a Complete Classification Theorem

This paper presents a unified algebraic--trigonometric framework for inverting Chebyshev polynomial maps. For each fixed integer n >= 2, the equation P_n(y) = C where P_n(y) = 2T_n(y/2) and T_n is the Chebyshev polynomial of the first kind admits a multi-valued analytic inversion via radicals: y = (A)^(1/n) + (B)^(1/n) with A and B defined explicitly in terms of C. A complete classification is provided: a polynomial admits such a radical inversion if and only if it is affinely conjugate to a Chebyshev polynomial. Connections to Dickson polynomials, dynamical semiconjugacy, and Galois theory are discussed.

This is a preprint that has not yet undergone peer review.