Timeline for Can a real quartic polynomial in two variables have more than 4 isolated local minima?
Current License: CC BY-SA 4.0
29 events
| when toggle format | what | by | license | comment | |
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| Mar 23, 2023 at 9:30 | comment | added | user44143 | Now we have the right answer — what a relief! | |
| Mar 21, 2023 at 11:56 | comment | added | Peter Mueller | @YaakovBaruch I guess edit #16 was the last one (unless some error requires a fix). I tried to put what was relevant or possibly interesting in the answer rather than spreading it out over the comments. In fact, I removed some obsolete comments. | |
| Mar 21, 2023 at 11:46 | comment | added | Yaakov Baruch | Excellent question and answer, but I think any further edits past #16 should be consigned to the comments. | |
| Mar 21, 2023 at 10:01 | history | edited | Peter Mueller | CC BY-SA 4.0 |
better example, I guess the last one ... :-)
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| Mar 20, 2023 at 23:41 | history | edited | Peter Mueller | CC BY-SA 4.0 |
One more example :-)
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| Mar 20, 2023 at 20:56 | history | edited | Peter Mueller | CC BY-SA 4.0 |
small adjustment of new example
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| Mar 20, 2023 at 19:33 | history | edited | Peter Mueller | CC BY-SA 4.0 |
Added another approach
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| Mar 20, 2023 at 10:11 | history | edited | Peter Mueller | CC BY-SA 4.0 |
added a link to Jap88's visualizations
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| Mar 19, 2023 at 9:47 | history | edited | Peter Mueller | CC BY-SA 4.0 |
added/modified sage code and added image for second example
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| Mar 18, 2023 at 23:31 | history | edited | Peter Mueller | CC BY-SA 4.0 |
one more rather short example
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| Mar 18, 2023 at 20:14 | history | edited | Peter Mueller | CC BY-SA 4.0 |
gave a better example
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| Mar 17, 2023 at 15:58 | history | edited | Peter Mueller | CC BY-SA 4.0 |
added a sparse example
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| Mar 17, 2023 at 12:35 | history | edited | Peter Mueller | CC BY-SA 4.0 |
incorporated some comments into the answer
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| Mar 17, 2023 at 3:53 | comment | added | Jap88 | By the way, I noticed that for this polynomial, when one traces the loci of $f_x=0$ and $f_y=0$ there is always one saddle point between two minima. | |
| Mar 17, 2023 at 3:39 | comment | added | Jap88 | I added a visualization of Peter Mueller's polynomial to the posting: math.stackexchange.com/questions/4620663 in case someone is interested. The polynomial is a bit tricky to visualize due to its "multiscale" nature. | |
| Mar 17, 2023 at 1:08 | comment | added | user44143 | @PeterMueller, with nice enough numbers those might make at least five minima obvious, but they would not show exactly five minima. By contrast, it is obvious that $(x^2-1)^2+(y^2-1)^2$ has exactly four local minima. | |
| Mar 16, 2023 at 21:26 | comment | added | user44143 | @DimaPasechnik, yes, $f+64$ is non-negative and therefore a sum of squares. What I want to know is: Can we write $f$ in a way which makes it obvious that it has exactly five local minima? | |
| Mar 16, 2023 at 21:05 | comment | added | Dima Pasechnik | (i.e., is it globally nonnegative? - then by Hilbert 1888 it will be sos) | |
| Mar 16, 2023 at 20:58 | comment | added | Dima Pasechnik | is this example an s.o.s. ? | |
| Mar 16, 2023 at 20:41 | history | edited | Peter Mueller | CC BY-SA 4.0 |
deleted 15 characters in body
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| Mar 16, 2023 at 17:46 | comment | added | Peter Mueller | @MattF. Very nice! I now found a similarly short example which in addition is symmetric in the variables. I adapted the answer. | |
| Mar 16, 2023 at 17:45 | history | edited | Peter Mueller | CC BY-SA 4.0 |
even better and symmetric example
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| Mar 16, 2023 at 17:34 | history | edited | Peter Mueller | CC BY-SA 4.0 |
smaller example
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| Mar 16, 2023 at 15:23 | comment | added | user44143 | Using your answer as a starting point, I found the simpler case $$f=26 x^2 - 472 x^3 + 341 x^4 + 624 x^2 y - 624 x^3 y + 156 y^2 + 5268 x y^2 + 960 x^2 y^2 - 8748 y^3 - 5268 x y^3 + 6483 y^4$$ which has local minima at $$(0,0),\ (0,1),\ (1,0),\ (3,2),\ (-2,-1)$$ | |
| Mar 16, 2023 at 14:17 | vote | accept | Jap88 | ||
| Mar 16, 2023 at 14:14 | comment | added | Jap88 | Great. I didn't believe such a polynomial existed. | |
| Mar 16, 2023 at 12:40 | history | edited | Peter Mueller | CC BY-SA 4.0 |
Original example replaced by simpler example
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| Mar 16, 2023 at 12:10 | history | edited | Peter Mueller | CC BY-SA 4.0 |
added positive example
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| Mar 15, 2023 at 22:14 | history | answered | Peter Mueller | CC BY-SA 4.0 |