Timeline for Topology of level sets for meromorphic function
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
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| Jul 6, 2023 at 16:44 | comment | added | kaleidoscop | What about the last formulation? | |
| Jul 5, 2023 at 17:32 | comment | added | Alexandre Eremenko | @Pierre PC: in either version, old or new, I do not see any well-defined question, to which a reasonable answer can be expected. | |
| Jul 5, 2023 at 9:34 | comment | added | Pierre PC | Of course I am not claiming that anything interesting can be said in this level of generality, but in this case the critic would be that the level of generality of the question is too high, and I am not sure the OP can be blamed for not being aware of this fact. Especially given that they don't seem to be sure that such functions even exist, and my opinion (maybe as a young and naïve researcher who does not know the answer himself) is that this is not something to be ashamed about. | |
| Jul 5, 2023 at 9:30 | comment | added | Pierre PC | @AlexandreEremenko I don't think the OP claims to have a definition of the natural probability distribution on all meromorphic functions. I think they just happen to define a random function that happens to be meromorphic on the plane and stationary. It is easy to construct such distributions, for instance by taking a random translate of a fixed elliptic function. I think the new question, although technically well-posed, is actually less illuminating: the initial question was more about if functions with unbounded gradient lines are in some sense "generic" or "rare". | |
| Jul 5, 2023 at 7:02 | comment | added | kaleidoscop | That is precisely the question. I put a binary question at the end if you think it is better to ask it like that. | |
| Jul 5, 2023 at 7:01 | history | edited | kaleidoscop | CC BY-SA 4.0 |
made the question binary
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| Jul 5, 2023 at 6:55 | comment | added | Alexandre Eremenko | What does it mean "to find conditions"? What sort of conditions? In terms of what? | |
| Jul 5, 2023 at 6:50 | comment | added | kaleidoscop | Ok :) I removed any reference to randomness. Any tool that says something about the topology of level lines of meromorphic functions can be of use to me. | |
| Jul 5, 2023 at 6:49 | history | edited | kaleidoscop | CC BY-SA 4.0 |
deleted 92 characters in body
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| Jul 5, 2023 at 6:22 | comment | added | Alexandre Eremenko | Then please restate your question so that it makes some precise meaning. | |
| Jul 5, 2023 at 4:35 | comment | added | kaleidoscop | Anyway thanks for the input, but randomnes was not really the point of my question anyway. | |
| Jul 5, 2023 at 4:30 | comment | added | kaleidoscop | Interesting. I would be curious to understand why $\sum_i \frac(1}{z-z_i}$ is not meromorphic. | |
| Jul 4, 2023 at 18:31 | comment | added | Alexandre Eremenko | The formula you wrote does not define a meromorphic function. Actually I consulted with one of the authors of that paper: he says that there is no reasonable definition of a random meromorphic function in the literature. | |
| Jul 4, 2023 at 7:28 | comment | added | kaleidoscop | Ok I should have been more precise. I am not talking about $f$, if you go p.3 of the paper, the potential $\nabla U=\nabla [\log(|f(z)|)-|z|^2/2]$ can be seen as a random meromorphic function of the previous form. I am not interested by this particular function but this is an example of a random meromorphic function. Thanks for your help. | |
| Jul 4, 2023 at 5:36 | review | Close votes | |||
| Jul 10, 2023 at 3:02 | |||||
| Jul 4, 2023 at 5:19 | comment | added | Alexandre Eremenko | In the paper you cite, the "random function" is ENTIRE, and of very special kind. No natural notion of randomness is known to me for MEROMORPHIC functions. | |
| Jul 3, 2023 at 19:37 | comment | added | kaleidoscop | I just talked about "random meromorphic functions" to help understand the kind of assumptions I am talking about. To keep it simple, consider random meromorphic functions of the form $$F(z)=\sum_i 1/(z-z_i)$$ where the set $\{z_i;i\geq 1\}$ is a random stationary set of points (the random sum is not always well defined, but it is well defined in some sense in some cases, see for instance link.springer.com/article/10.1007/s00039-007-0613-z) | |
| Jul 3, 2023 at 16:53 | comment | added | Alexandre Eremenko | Can you explain exactly what is a "random and stationary meromorphic function"? What is your definition of a probability measure on the set of all meromorphic functions in C? | |
| Jul 3, 2023 at 15:18 | history | asked | kaleidoscop | CC BY-SA 4.0 |