Timeline for A game on Noetherian rings
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
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| May 19, 2023 at 4:54 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals from title
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| May 19, 2023 at 1:01 | answer | added | Chris Gerig | timeline score: 12 | |
| May 5, 2017 at 8:15 | comment | added | Martin Brandenburg | @Joël: I have written a GAP-program which helps to analyse the game for finite rings. One has to input the initial ring using the method RingByStructureConstants. | |
| Aug 20, 2014 at 16:39 | comment | added | Gabriel C. Drummond-Cole | This is also similar to Sylver Coinage. | |
| May 9, 2012 at 19:47 | comment | added | Will Sawin | The ring is a quotient of a Dedekind domain, thus, a product of quotients of DVRs. Choose the maximal ideal from one DVR and the unit ideal from the others. Also: In my Hilbert irreducibility argument, I neglected to mention the obviously critical assumption that the original polynomial is irreducible. Playing something reducible, like x(x-1), forces them to pass you a finite-dimension non-field (with a principal maximal ideal.) | |
| May 9, 2012 at 19:36 | comment | added | Martin Brandenburg | @Will: Thanks; but I don't understand the argument. I don't see why every ring of finite vector space dimension $\geq 2$ over $k$ can be moved to some field immediately, i.e. that it has some principal maximal ideal. | |
| May 4, 2012 at 19:58 | comment | added | Will Sawin | In fact, Hilbert irreducibility implies that this is true. | |
| May 4, 2012 at 19:56 | comment | added | Will Sawin | If $k$ is algebraically closed, the player who plays on it wins. Any Weierstrauss equation will do, since the opponent must pass you a ring of finite $\geq 2$ dimension over $k$, with which you can always pass them a field. If it's not algebraically closed, this might not work, and I'm not sure what to do. My guess is that if $k$ is a number field then the first player to play loses. | |
| May 4, 2012 at 14:25 | comment | added | Martin Brandenburg | Does anyone have a idea if $k[x,y]$ wins or loses? | |
| Apr 21, 2012 at 17:05 | answer | added | Martin Brandenburg | timeline score: 35 | |
| Apr 7, 2012 at 11:15 | answer | added | Kevin Buzzard | timeline score: 45 | |
| Apr 6, 2012 at 18:56 | comment | added | Joël | Question: are the algebra softwares strong enough so that it would possible to actually implement this game ? If so, we could organized a tournament. That would be really fun. | |
| Apr 6, 2012 at 12:56 | answer | added | Pat Devlin | timeline score: 12 | |
| Apr 6, 2012 at 12:00 | comment | added | Filippo Alberto Edoardo | @Jerôme: I absolutely agree with you, I was reporting on a bad idea... | |
| Apr 6, 2012 at 11:40 | comment | added | Malte | I am absolutely shocked no one has pointed out the similarities to Choquet's game. Baire's theorem gives a characterization of the existence of a winning strategy for one of the players: www.math.auckland.ac.nz/~moors/game.pdf Question: Why not use limites to extend the game to any type of ring? | |
| Apr 6, 2012 at 10:37 | comment | added | Jérôme Poineau | @FAE: I don't believe that Krull dimension is the right thing to consider. In a geometric situation like that of the OP, the minimal dimension of an irreducible component should play a role (you can win in one move if you have the union of a line and a plane for instance). And there are reduction issues too (start with a double line). | |
| Apr 6, 2012 at 9:26 | history | edited | Martin Brandenburg | CC BY-SA 3.0 |
added 9 characters in body
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| Apr 6, 2012 at 9:12 | comment | added | Filippo Alberto Edoardo | Uhm..nice! I hoped to reduce to regular rings, so the winning position would coincide with the value $\pmod{2}$ of the Krull dimension. But since quotients of regular rings may fail to be regular again (see $\mathbb{Z}/p^2$), it leads nowhere... | |
| Apr 6, 2012 at 8:06 | comment | added | Olivier | Does it tell too much about me that I think this game would be kind of fun? | |
| Apr 6, 2012 at 3:24 | history | asked | Will Sawin | CC BY-SA 3.0 |