What question did this study set out to answer?

The aim is to develop a method for inverting Chebyshev polynomial maps using radicals and classify the conditions required for such inversions.

April 14, 2026Open Access

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

Key Points

The aim is to develop a method for inverting Chebyshev polynomial maps using radicals and classify the conditions required for such inversions.
Established a multi-valued analytic inversion equation for Chebyshev polynomials.
Defined variables A and B explicitly in terms of a constant C.
Classified polynomials based on their relationship to Chebyshev polynomials.
A radical inversion is possible if the polynomial is affinely conjugate to a Chebyshev polynomial.
Presented connections to Dickson polynomials and dynamical semiconjugacy.

Abstract

This paper presents a unified algebraic--trigonometric framework for inverting Chebyshev polynomial maps. For each fixed integer n >= 2, the equation Pₙ (y) = C where Pₙ (y) = 2Tₙ (y/2) and Tₙ 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.

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

Key Points

Abstract

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