Timeline for Starshapeness of polynomial tracts with respect to the (entire collection of) critical points contained in the tract
Current License: CC BY-SA 3.0
5 events
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| Dec 6, 2014 at 22:50 | comment | added | Trevor J Richards | (continued) and thus of course the bounded face of $A$ must contain many zeros of $p'$ if it contains many zeros of $p$. | |
| Dec 4, 2014 at 15:41 | comment | added | Trevor J Richards | Moreover, it seems that if the Jordan Curve $J$ that the lemniscate $\Lambda=\{z:|p(z)|=\epsilon\}$ approximates is very complicated, then the bounded face of $\Lambda$ may need to contain many zeros of $p$ for $\Lambda$ to approximate $J$ well. | |
| Dec 3, 2014 at 15:06 | comment | added | Trevor J Richards | Ah, but I do not need it to be starshaped with respect to just one point, I want it to be starshaped with respect the family of critical points. Thus, each point in the tract can be "seen" by some critical point. I will make that more clear in the question. | |
| Dec 3, 2014 at 0:18 | comment | added | Malik Younsi | There is also a simple proof using potential theory in the book "Potential theory in the complex plane" by Thomas Ransford, Theorem 5.5.8. I don't have access to Walsh's book right now, so I don't know if the proof is the same. | |
| Dec 2, 2014 at 21:11 | history | answered | Alexandre Eremenko | CC BY-SA 3.0 |