Timeline for How many strict local minima a quartic polynomial in two variables might have?
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| May 15, 2023 at 14:34 | vote | accept | Pavel Kocourek | ||
| Mar 7, 2023 at 4:06 | answer | added | Jap88 | timeline score: 8 | |
| S Feb 3, 2023 at 15:02 | history | bounty ended | CommunityBot | ||
| S Feb 3, 2023 at 15:02 | history | notice removed | CommunityBot | ||
| Jan 31, 2023 at 16:38 | comment | added | Pavel Kocourek | @HagenvonEitzen Thanks! Corrected. | |
| Jan 31, 2023 at 16:37 | history | edited | Pavel Kocourek | CC BY-SA 4.0 |
deleted 4 characters in body
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| Jan 31, 2023 at 5:27 | comment | added | Hagen von Eitzen | Typo in the “at most 9“ part: you mean cubic, not quadratic (cf. the referenced comment) | |
| Jan 29, 2023 at 21:00 | history | tweeted | twitter.com/StackMath/status/1619802584033771521 | ||
| Jan 27, 2023 at 13:41 | comment | added | Alex K | I think for $p$ where $p\rightarrow \infty$ in all directions, you should get critical points in addition to minima. If you consider two minima, and a curve that passes between them, we know the derivative of $p$ changes sign along that curve... At least, it makes the approach in that linked answer no longer work. | |
| S Jan 26, 2023 at 13:46 | history | bounty started | Pavel Kocourek | ||
| S Jan 26, 2023 at 13:46 | history | notice added | Pavel Kocourek | Draw attention | |
| Jan 19, 2023 at 3:45 | history | edited | Pavel Kocourek | CC BY-SA 4.0 |
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| Jan 18, 2023 at 4:50 | comment | added | Pavel Kocourek | @GerryMyerson Great thanks for this observation! On the real line the roots of $f'$, if all being simple roots, are alternatively local minima and local maxima. This question gives an example of a function that has two minima and no other critical point: math.stackexchange.com/q/4024737/1134951. Finding a function with 9 local minima and no maximum would perhaps be harder. | |
| Jan 18, 2023 at 3:37 | comment | added | Gerry Myerson | Here's some rough-and-ready reasoning that suggests an answer and an approach (without actually proving anything). We expect that at isolated local extreme points both partial derivatives vanish. The partial derivatives are cubics. By Bezout's Theorem, we expect the two cubics to have $3\times3=9$ common zeros. | |
| Jan 18, 2023 at 2:03 | history | edited | Pavel Kocourek | CC BY-SA 4.0 |
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| Jan 18, 2023 at 1:55 | history | asked | Pavel Kocourek | CC BY-SA 4.0 |