Timeline for Are there any known criteria for quadratic mapping from R^n to R^n being surjective?
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Jan 2, 2014 at 14:33 | comment | added | Rampant_mouse | y1 = 2 * x1 * x3; y2 = 2 * x2 * x3; y3 = x3^2 - x1^2 - x2^2. To see it is enough to write it in polar coordinates. For higher dimensions the situation is same yj = 2 * xj * xn; yn = xn^2 - squares of other variables | |
| Jan 2, 2014 at 14:07 | comment | added | Rampant_mouse | It is easy to see that the surjective quadratic mappings exist in every dimension. For example in dimension 3 it is the following mapping: | |
| Dec 31, 2013 at 15:36 | comment | added | joro | @abx I suspect invertible quadratic map exist with constant det(jacobian) and explicit polynomial inverse. | |
| Dec 31, 2013 at 15:34 | comment | added | abx | Yes (a quadratic map). | |
| Dec 31, 2013 at 15:32 | comment | added | joro | @abx Are you asking if a surjective map exists for odd n >= 3 ? | |
| Dec 31, 2013 at 15:13 | comment | added | abx | Still I am curious to know whether the map can be surjective for $n$ odd $\geq 3$? | |
| Dec 31, 2013 at 12:02 | comment | added | Pietro Majer | This is a nice remark, but it does not settle the case of even dimension. The OP asks for a criterium... | |
| Dec 31, 2013 at 9:09 | history | edited | Glasby | CC BY-SA 3.0 |
Clarify the generalization to all even $n$
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| Dec 31, 2013 at 8:59 | history | answered | Glasby | CC BY-SA 3.0 |