Polynomials invertible in k-radicals

TL;DR

This paper extends Ritt's classical result by characterizing polynomials invertible in k-radicals, showing they are compositions of specific polynomial types, including exceptional cases with special monodromy groups for k<15.

Contribution

It generalizes the classification of invertible polynomials in radicals to include solutions of equations up to degree k, incorporating exceptional polynomials with specific monodromy groups.

Findings

Polynomials invertible in radicals are compositions of power, Chebyshev, and degree ≤4 polynomials.

For k<15, certain exceptional polynomials with special monodromy groups are included.

The classification relies on group-theoretical analysis of monodromy groups.

Abstract

A classic result of Ritt describes polynomials invertible in radicals: they are compositions of power polynomials, Chebyshev polynomials and polynomials of degree at most 4. In this paper we prove that a polynomial invertible in radicals and solutions of equations of degree at most k is a composition of power polynomials, Chebyshev polynomials, polynomials of degree at most k and, if k < 15, certain polynomials with exceptional monodromy groups. A description of these exceptional polynomials is given. The proofs rely on classification of monodromy groups of primitive polynomials obtained by M\"{u}ller based on group-theoretical results of Feit and on previous work on primitive polynomials with exceptional monodromy groups by many authors.