In the recent paper 

<cite authors="Anton, R.; Mihalache, N.; Vigneron, F.">_Anton, R.; Mihalache, N.; Vigneron, F._, [**A Short ODE Proof of the Fundamental Theorem of Algebra.**](https://doi.org/10.1007/s00283-023-10308-z), Math. Intelligencer (2023).</cite>

the authors give a proof based on the analysis of the attractive points of the initial value problem:

$$\varphi_z(0)=z, \quad \varphi_z^{\prime}(t)=-\frac{P\left(\varphi_z(t)\right)}{P^{\prime}\left(\varphi_z(t)\right)},$$


which is a continuous-time version of Newton's method for computing the zeros of the polynomial $P$.