Timeline for Continuous map from $\mathbb C^4$ to $\mathbb R$ that changes sign under circular permutation of coordinates and that is $0$ only for squares
Current License: CC BY-SA 4.0
23 events
| when toggle format | what | by | license | comment | |
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| Apr 21, 2020 at 15:09 | comment | added | jcdornano | The disproof has been given by the great Anatole Khelif ! Anatole just answered now to the question "do you mind if I mention your name?" in a contrast, he took just one minut to answer by mail and by the negative, the question of the post! | |
| Apr 17, 2020 at 11:05 | comment | added | jcdornano | The idea is that if $abcd$ is not a square, one can find a path from it to $bcda$ such that no 4-gon on the path is a square. This is a contradiction because as require in the hypothesis $f(a,b,c,d)$ and $f(b,c,d,a)$ have a different sign, so $f$ continuous has to take the value $0$ for some 4-gone on the path, and this can happend only if the 4-gon is a square. | |
| Apr 17, 2020 at 9:35 | comment | added | jcdornano | i don't inderstand the upvotes of the comments, especially the one of Gerry Myerson, that does not give an example, because as I said it just after, f is supposed to be real valued...Actually the question can be answer by the negative, and I'm going to explain why in an answer, in a moment. | |
| Apr 17, 2020 at 2:22 | comment | added | jcdornano | @LSpice, thank you very much for the english grammar and presentation edit ! | |
| Apr 17, 2020 at 1:55 | history | edited | LSpice | CC BY-SA 4.0 |
Proofreading
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| Apr 17, 2020 at 1:24 | comment | added | jcdornano | I saved the question of uselessity in case of an eventual trivial positive answer by an edit at the end, that take in consideration and give an issue to what I wrote in my upper comment. I also removed the tag model theory | |
| Apr 17, 2020 at 1:14 | history | edited | jcdornano | CC BY-SA 4.0 |
added 99 characters in body; edited tags
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| Apr 17, 2020 at 0:53 | comment | added | jcdornano | I realized that I didn't post what I wanted, and that I previously switched quantificator in the i) , i'm going to ask a new question, by replacing "there exists" by "for all" ... or for an in between question, replace there exists in i) by for all , but with some condition on the type of thz polygone xyzt , this condition should be that there is a polygone of this type in any Jordan curve (why not triangles...) | |
| Apr 17, 2020 at 0:46 | comment | added | jcdornano | @Gerry Myerson $f$ is real valued. | |
| Apr 17, 2020 at 0:43 | history | edited | jcdornano | CC BY-SA 4.0 |
deleted 104 characters in body
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| Apr 17, 2020 at 0:24 | comment | added | jcdornano | Thank you Joseph O'Rourke, i mention this problem in the question but i now have a link ... that i m going to open right now , and add the in the question! | |
| Apr 17, 2020 at 0:19 | comment | added | jcdornano | @ Joseph O'Rourke, we almost posted at the same time. that is why i don't mension your comment, but endeed, you had the right interoretation... maybe I should edit the main post to avoid misunderstanding... | |
| Apr 17, 2020 at 0:16 | comment | added | Joseph O'Rourke | See the Inscribed Square Problem: "Does every plane simple closed curve contain all four vertices of some square?" | |
| Apr 17, 2020 at 0:14 | comment | added | jcdornano | i mean they are the vertices of a square i.e. they satisfy $|x-y| = |y-z| = |z-t|= |t-x| $ and $|x-z|=|y-t|$. I used $\mathbb C$ instead of $\mathbb R^2$ for presentation, but same equalities hold by replacing $|a+bi|$ by $a^2+b^2$ , so that the problem might be enounced in first order in some more general context. | |
| Apr 17, 2020 at 0:14 | comment | added | Joseph O'Rourke | @GerryMyerson: Perhaps those four complex numbers form the corners of a square in the complex plane? | |
| Apr 16, 2020 at 23:10 | comment | added | Gerry Myerson | Not every question about polynomials is a question about model theory. But what does it mean for $(x,y,z,t)$ to be a square? | |
| Apr 16, 2020 at 21:59 | comment | added | jcdornano | The problem if f is a polynome is related to a first order formula in the theory of ordered fields, isn't it? | |
| Apr 16, 2020 at 21:52 | comment | added | Wojowu | Whatever definissable sets are, I do not thing the tag is appropriate. | |
| Apr 16, 2020 at 21:50 | comment | added | jcdornano | I guess for definissable sets, do you think i should remove this tag? | |
| Apr 16, 2020 at 21:39 | history | edited | jcdornano | CC BY-SA 4.0 |
edited title
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| Apr 16, 2020 at 21:38 | comment | added | Wojowu | How does $f$ being a polynomial ("polynome") justify the tag "model theory"? | |
| Apr 16, 2020 at 21:34 | history | edited | jcdornano | CC BY-SA 4.0 |
edited title
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| Apr 16, 2020 at 21:17 | history | asked | jcdornano | CC BY-SA 4.0 |