Completely solving the quintic by iteration

TL;DR

This paper presents a novel iterative method that leverages icosahedral symmetry to completely solve the quintic polynomial in a single dynamical process, advancing algebraic and geometric solution techniques.

Contribution

It introduces a new map that breaks all quintic symmetry, enabling the extraction of all roots in one iteration, building on Klein's and Doyle & McMullen's symmetry-based approaches.

Findings

A new dynamical map for the quintic is constructed.

The method allows for all roots to be found in a single iteration.

The approach unifies algebraic and geometric perspectives on solving the quintic.

Abstract

In the late nineteenth century, Felix Klein revived the problem of solving the quintic equation from the moribund state into which Galois had placed it. Klein's approach was a mix of algebra and geometry built on the structure of the regular icosahedron. His method's key feature is the connection between the quintic's Galois group and the rotational symmetries of the icosahedron. Roughly a century after Klein's work, P. Doyle and C. McMullen developed an algorithm for solving the quintic that also exploited icosahedral symmetry. Their innovation was to employ a symmetrical dynamical system in one complex variable. In effect, the dynamical behavior provides for a partial breaking of the polynomial's symmetry and the extraction of two roots following one iterative run of the map. The recent discovery of a map whose dynamics breaks all of the quintic's symmetry allows for five roots to emerge from a single run. After sketching some algebraic and geometric background, the discussion works out an explicit procedure for solving the quintic in a complete sense.