Explicitly solvable algebraic equations of degree 8 and 9

TL;DR

This paper demonstrates explicit radical solutions for degree 8 and 9 polynomials when coefficients are constrained by 6 parameters, revealing new solvability conditions and parameterizations.

Contribution

It introduces a method to explicitly solve degree 8 and 9 polynomials using 6 parameters, under specific coefficient constraints.

Findings

Explicit radical solutions for degree 8 and 9 polynomials.

Constraints on coefficients derived from parameterization.

Explicit formulas for parameters in terms of coefficients.

Abstract

The generic monic polynomial of degree N features N a priori arbitrary coefficients $c_m$ and N zeros $z_n$. In this paper we limit consideration to $N = 8$ and $N = 9$. We show that if the $N$ -- a priori arbitrary -- coefficients $c_m$ of these polynomials are appropriately defined -- as it were, a posteriori -- in terms of 6 arbitrary parameters, then the $N$ roots of these polynomials can be explicitly computed in terms of radicals of these 6 parameters. We also report the constraints on the N coefficients $c_m$ implied by the fact that they are so defined in terms of 6 arbitrary parameters; as well as the explicit determination of these 6 parameters in terms of the N coefficients $c_m$.