Timeline for Can discriminant polynomials become perfect powers on hyperplanes?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 9, 2018 at 18:58 | vote | accept | Stanley Yao Xiao | ||
| Nov 9, 2018 at 1:20 | comment | added | Stanley Yao Xiao | Let us continue this discussion in chat. | |
| Nov 8, 2018 at 16:35 | comment | added | Libli | If you go to any Grassmannians, the duality is much harder to formulate (Chow forms) and has less powerful consequences. But the Chow forms have been studied by many people. You may first have a look to the book "Discriminants, resultants and multidimnesional determinants" by Gelfand-Kapranov-Zelevinsly. | |
| Nov 8, 2018 at 16:33 | comment | added | Libli | The question could be "what is the locus, in the appropriate Grassmannians, such that the restriction of the discriminant is a square on each linear spaces parametrized by this locus?" This seems to be a much more difficult question. Indeed, the theory of projective duality is quite rich, precisely because there is a duality. | |
| Nov 8, 2018 at 16:31 | comment | added | Libli | I don't know if this second question is well-posed. If you take different lines, the system of coordinates vary with the lines you take and this doesn't look obvious how to relate one square to the other. | |
| Nov 8, 2018 at 15:24 | comment | added | Stanley Yao Xiao | For example, when $d = 4$ the locus contains a countable union of lines. An example of such a line is given by $a_3 = a_1 = 0, a_4 = a_0$, giving rise to polynomials of the shape $a_4 x^4 + a_2 x^2 + a_4$. The discriminant is identically a square on this line. | |
| Nov 8, 2018 at 15:22 | comment | added | Stanley Yao Xiao | the square is actually just a square, i.e., I want to look for the locus on which $\Delta_d(f)$ is equal to the square of some polynomial | |
| Nov 8, 2018 at 14:41 | comment | added | Libli | can't read the square. You want to describe the locus such that $\Delta_d(f) = ???$? | |
| Nov 8, 2018 at 14:12 | comment | added | Stanley Yao Xiao | What you wrote here is really interesting... is it possible to use this to describe the locus, for a given $d$, such that $\Delta_d(f) = \square$? | |
| Nov 7, 2018 at 22:17 | history | answered | Libli | CC BY-SA 4.0 |