Timeline for Refinement of mean value conjecture for complex polynomials?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 2, 2021 at 17:11 | vote | accept | Stefan Steinerberger | ||
| Jun 1, 2021 at 11:53 | history | edited | Peter Mueller | CC BY-SA 4.0 |
Added another example
|
| May 31, 2021 at 23:20 | history | edited | Peter Mueller | CC BY-SA 4.0 |
More examples and simpler proof
|
| May 28, 2021 at 5:38 | history | edited | Peter Mueller | CC BY-SA 4.0 |
Added simpler counterexample
|
| May 27, 2021 at 18:04 | comment | added | Stefan Steinerberger | Thanks for the details (and, of course, the construction)! | |
| May 27, 2021 at 17:46 | comment | added | Peter Mueller | As to integer coordinates: First I computed in the complex floats, and in the first example found, I chose approximations of the coefficients in $\mathbb Q(i)$ which are sufficiently good so that the example remains an example. | |
| May 27, 2021 at 17:10 | comment | added | Peter Mueller | I did nothing real smart, in fact it was a search for random polynomials (with random roots, and also with random coefficients). This one was the 3403th tested case in degree $5$. I ran various degrees up to $30$ in parallel. Starting in $0$ I constructed a polygon which approximates the border of $B$, and used the argument principle along the polygon to count the roots of $g(z)$ and $g(z)+zg'(z)$, respectively. (There seem to be more examples, but haven't checked them.) | |
| May 27, 2021 at 16:53 | comment | added | Stefan Steinerberger | I first thought that since all the roots have integer coordinates, the example must be somewhat robust -- after playing with it a little, it seems like it is actually fairly delicate. How was it constructed? | |
| May 27, 2021 at 16:48 | comment | added | Peter Mueller | Thanks, I fixed the typo in the last root. | |
| May 27, 2021 at 16:46 | history | edited | Peter Mueller | CC BY-SA 4.0 |
edited body
|
| May 27, 2021 at 16:36 | history | bounty ended | Stefan Steinerberger | ||
| May 27, 2021 at 16:36 | comment | added | Stefan Steinerberger | Amazing!! I think there is a sign flip in the last root (-45 + 18i instead of -45 - 18i) but the picture shows it very clearly and I was able to reproduce the example numerically (I get that $g(\mbox{critical point}) = 1.001...$ for the critical point in question in question). | |
| May 27, 2021 at 16:25 | history | edited | Peter Mueller | CC BY-SA 4.0 |
deleted 150 characters in body
|
| May 27, 2021 at 16:06 | history | edited | Peter Mueller | CC BY-SA 4.0 |
Added sage code
|
| May 27, 2021 at 13:59 | history | edited | Peter Mueller | CC BY-SA 4.0 |
Modified example with easier coefficients
|
| May 27, 2021 at 11:45 | history | answered | Peter Mueller | CC BY-SA 4.0 |