Timeline for Transformations of the cubic forms [closed]
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Sep 4, 2019 at 16:27 | comment | added | Dima Pasechnik | the next step, if this didn't work, would be to ask @alex-degtyarev :-) | |
| Sep 4, 2019 at 16:25 | comment | added | Dima Pasechnik | a necessary condition would be, for two smooth surfaces, to check their Schlaefli type: pdfs.semanticscholar.org/ea71/… | |
| Sep 4, 2019 at 16:19 | comment | added | Dima Pasechnik | @darijgrinberg - of course, non-singularity is important. By the way, perhaps one can do the classical (or geometric?) invariant theory thing here, although the generators of the ring of invariants of 4-ary cubics aren't known IIRC. | |
| Sep 4, 2019 at 16:12 | comment | added | abx | $F_2$ is divisible by $x_1$. | |
| Sep 4, 2019 at 16:08 | comment | added | Anton Nedelin | @Dima Pasechnik: Thank you. That was my first idea. I would only say that there is $4\times 4$ matrix of coefficients in this case. However trying to solve particular example above in Mathematica was not very successful. So I was wondering if the transformation exists at all and if there are more effective algorithms known. | |
| Sep 4, 2019 at 16:05 | comment | added | Anton Nedelin | @abx: Thank you for the suggestion. I was looking for some criteria like this. However could you please elaborate a bit. I would think that $F_2=0$ has node singularities at $(1,0,0,0)$ and its permutations just like $F_1=0$ has. Why is this not a case? | |
| Sep 4, 2019 at 15:25 | comment | added | darij grinberg | @DimaPasechnik: To be fully precise, you would have to also encode the nonsingularity of the transformation matrix as a polynomial inequation. While there are algorithms for this using Grobner bases (you have to check whether a polynomial vanishes all over the zero set of an ideal), the question whether there are better algorithms than that is reasonable. Still, probably math.stackexchange material. | |
| Sep 4, 2019 at 15:24 | history | closed |
Dima Pasechnik abx Alexey Ustinov Piotr Hajlasz darij grinberg |
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| Sep 4, 2019 at 9:25 | review | Close votes | |||
| Sep 4, 2019 at 15:25 | |||||
| Sep 4, 2019 at 9:21 | comment | added | abx | In your example the answer is no. The surface $F_1=0$ has isolated singularities, while $F_2=0$ is reducible. | |
| Sep 4, 2019 at 9:09 | history | edited | YCor |
edited tags; edited tags
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| Sep 4, 2019 at 9:07 | comment | added | YCor | You need to understand the orbits of the action of $\mathrm{GL}(4,K)$ on $S^3(K^4)$. For such questions you need to be more specific about the ground field. | |
| Sep 4, 2019 at 9:06 | comment | added | Dima Pasechnik | surely, there is an algorithm reducing this question to solving a system of polynomial equations with unknowns being the entries of a 3x3 matrix. | |
| Sep 4, 2019 at 9:05 | review | First posts | |||
| Sep 4, 2019 at 11:09 | |||||
| Sep 4, 2019 at 9:02 | history | asked | Anton Nedelin | CC BY-SA 4.0 |