Timeline for When are the level curves of a polynomial bounded?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 25, 2022 at 15:20 | comment | added | YCor | @AlexandreEremenko I know, but the question specifies "all sufficiently large (positive) $k$" | |
| Nov 25, 2022 at 14:06 | comment | added | Alexandre Eremenko | @YCor: I disagree: for a constant polynomial, one level set is unbounded. | |
| Nov 25, 2022 at 10:25 | comment | added | YCor | For $n\ge 2$ and a continuous map $f:\mathbf{R}^n\to\mathbf{R}$, it is easy to check that $f^{-1}(\{k\})$ is bounded for all $k$ large enough, if and only if either $f$ is bounded or $|f(x)|\to\infty$ when $\|x\|\to\infty$. | |
| Nov 25, 2022 at 10:23 | comment | added | YCor | @AlexandreEremenko Well, constant maps also satisfy the condition. | |
| Nov 25, 2022 at 10:20 | comment | added | YCor | For a Zariski closed subset $X$ of $\mathbf{R}^2$, the property "$X$ consists of a finite number of bounded components" is equivalent to "$X$ is bounded". So you require that for every $k$ (real? integer?) large enough, $f^{-1}(\{k\})$ is bounded. Or I misunderstand your formulation? | |
| Nov 25, 2022 at 2:55 | comment | added | Alexandre Eremenko | The necessary and sufficient condition is that $|f(x,y)|\to+\infty$ as $x^2+y^2\to +\infty$. | |
| Nov 25, 2022 at 2:37 | history | asked | Stanley Yao Xiao | CC BY-SA 4.0 |