Solving polynomials with ordinary differential equations

TL;DR

This paper demonstrates that roots of polynomial families can be characterized as solutions to specific ordinary differential equations, providing a unified approach to solving polynomials of degree up to four.

Contribution

It introduces a simple, self-contained method linking polynomial roots to various ODEs, including Abel and linear ODEs, for degrees up to four.

Findings

Roots satisfy separated variables ODEs

Roots satisfy a first order generalized Abel ODE

Roots satisfy an (n-1)-th order linear ODE

Abstract

In this work we consider a given root of a family of n-degree polynomials as a one-variable function that depends only on the independent term. Then we prove that this function satisfies several ordinary differential equations (ODE). More concretely, it satisfies several simple separated variables ODE, a first order generalized Abel ODE of degree n-1 and an (n-1)-th order linear ODE. Although some of our results are not new, our approach is simple and self-contained. For n=2, 3 and 4 we recover, from these ODE, the classical formulas for solving these polynomials.