The Gauss-Lucas Theorem
Complex Variables
2021-12-02 v1
Abstract
The Gauss-Lucas theorem says that for any complex polynomial , the roots of the derivative lie in the convex hull of the roots of . In other words, the roots of lie inside the smallest convex subset of the complex plane containing all the roots of . 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
Cite
@article{arxiv.2112.00110,
title = {The Gauss-Lucas Theorem},
author = {John C. Baez},
journal= {arXiv preprint arXiv:2112.00110},
year = {2021}
}
Comments
2 pages, one picture by Greg Egan