Timeline for Which of the proofs of the fundamental theorem of algebra can actually produce bounds on where the roots are?
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| Oct 15, 2019 at 8:45 | answer | added | Mare | timeline score: 8 | |
| Oct 9, 2019 at 19:09 | comment | added | user44143 | Similarly the proof via Liouville's theorem, e.g. the first paragraph at en.wikipedia.org/wiki/… and then the paragraph called "another analytic proof", shows that if $|p(z)|>|p(0)|$ whenever $|z|>r$, then there is a root $z_0$ with $|z_0|<r$. Wikipedia's proof has a gap in that it doesn't show you how to compute this $r$, but once again $\max(1,\sum|a_i|)$ is good enough. | |
| Sep 24, 2019 at 19:46 | comment | added | Will Sawin | I think the fundamental group argument also essentially produces roots in the same disk as the Rouche's theorem one - we want to find a circle in the complex plane that wraps $n$ times around $0$, for which it suffices to know the leading term dominates as then we can deform one to the other without touching zero. I guess this is more or less how Rouche's theorem is proved. | |
| Sep 21, 2019 at 16:32 | history | asked | Qiaochu Yuan | CC BY-SA 4.0 |