Timeline for How many possible values for the determinant of an $n\times n$-matrix with entries $1,2,\dots,n^2$?
Current License: CC BY-SA 4.0
6 events
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| Jun 7, 2021 at 19:58 | comment | added | François Brunault | Using the techniques in this MO answer, it is possible to compute the expected value of $\det(A)^2$ where $A$ is a $n \times n$-matrix whose coefficients are independent random variables in $\{1,\ldots,n^2\}$. Your setting is different in that the coefficients are all distinct, but it gives an idea of the size of the determinant. | |
| Jun 7, 2021 at 18:30 | comment | added | Moritz Firsching | @Wojowu I agree that the asymptotics would be even more interesting; don't know anything non-obvious about it though.. | |
| Jun 7, 2021 at 16:41 | comment | added | Nathaniel Johnston | Gaspar's determinant theorem can be used to further reduce this bound to something like $O(n^{5n/2})$, which appears to have at least roughly the right growth rate. | |
| Jun 7, 2021 at 16:01 | comment | added | Peter Taylor | @Wojowu, permuting rows or columns only changes the sign of the determinant, so the obvious bound is really $2(n^2)!(n!)^{-2}$ (although the point that the values are considerably smaller still stands) | |
| Jun 7, 2021 at 15:37 | comment | added | Wojowu | I personally would be more interested in its asymptotics than the complexity of computing it. Do you happen to know anything about that? It is significantly smaller than the most obvious bound of $(n^2)!$. | |
| Jun 7, 2021 at 15:22 | history | answered | Moritz Firsching | CC BY-SA 4.0 |