The Gauss-Lucas Theorem

TL;DR

The paper explains the Gauss-Lucas theorem, which states that the roots of a polynomial's derivative lie within the convex hull of the original roots, using an electrostatic analogy for intuitive understanding.

Contribution

It provides an intuitive electrostatic explanation for the Gauss-Lucas theorem, enhancing conceptual understanding of the roots' behavior.

Findings

Electrostatic analogy clarifies root locations

Roots of derivative lie within convex hull

Provides intuitive proof of the theorem

Abstract

The Gauss-Lucas theorem says that for any complex polynomial $P$, the roots of the derivative $P'$ lie in the convex hull of the roots of $P$. In other words, the roots of $P'$ lie inside the smallest convex subset of the complex plane containing all the roots of $P$. This theorem is not hard to prove, but is there an intuitive explanation? In fact there is, using physics -- or more precisely, electrostatics in 2-dimensional space