Timeline for Can a variety be the graph of a function in more than one way?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Oct 15 at 21:11 | comment | added | M. Winter | Why the downvote? | |
| Oct 15 at 14:25 | answer | added | M. Winter | timeline score: 0 | |
| Oct 5 at 23:03 | vote | accept | M. Winter | ||
| Oct 4 at 19:06 | vote | accept | M. Winter | ||
| Oct 4 at 19:06 | |||||
| Oct 4 at 15:00 | answer | added | Will Sawin | timeline score: 9 | |
| Oct 2 at 18:29 | comment | added | M. Winter | @WillSawin Even though I can think of ways to modify my question to avoid such examples, I also think your example was sufficiently unexpected for me that I would accept it as an answer. | |
| Oct 2 at 14:51 | comment | added | Will Sawin | Also the set of $x,y,z$ such that $\frac{1}{x}+ \frac{1}{y} + \frac{1}{z}=0$ can be expressed in three ways as the graph of a function since each of $x,y,z$ may be expressed as a rational function in the other two. | |
| Oct 2 at 14:45 | comment | added | Andy Putman | @M.Winter: There are many invertible rational maps. In fact, there are even many invertible polynomial maps (see en.wikipedia.org/wiki/Jacobian_conjecture). | |
| Oct 2 at 14:35 | history | edited | M. Winter | CC BY-SA 4.0 |
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| Oct 2 at 14:29 | comment | added | M. Winter | @SamHopkins Good point as well. My question would then be whether swapping $X$ and $Y$ can give the only other solutions. And are fractional linear functions the only rational functions with rational inverse? Otherwise any other such function would allow for the same trick. | |
| Oct 2 at 14:00 | comment | added | Sam Hopkins | What if $f$ is invertible, like a fractional linear transformation? Then can't we swap the role of $X$ and $Y$? | |
| Oct 2 at 13:55 | history | edited | M. Winter | CC BY-SA 4.0 |
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| Oct 2 at 13:43 | history | edited | M. Winter | CC BY-SA 4.0 |
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| Oct 2 at 13:41 | comment | added | M. Winter | @JasonStarr Good point. I need my decomposition $X\oplus Y$ to be orthogonal as otherwise a transformation as yours always yields counterexamples. I made an edit. | |
| Oct 2 at 12:27 | comment | added | Jason Starr | The graph of $y=x^2$ is also the graph of $z=x^2 + ax$ for the coordinate $z=y+ax$. | |
| Oct 2 at 11:48 | history | asked | M. Winter | CC BY-SA 4.0 |