The Gauss-Lucas Theorem

John C. Baez

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arXiv · 2112.00110

The Gauss-Lucas Theorem

John C. Baez

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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

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John C. Baez. 2021-11-30. The Gauss-Lucas Theorem. https://arxiv.org/abs/2112.00110

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