Timeline for Number of unique determinants for an NxN (0,1)-matrix
Current License: CC BY-SA 3.0
19 events
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| Feb 28, 2017 at 17:44 | comment | added | Will Orrick | …seem skeptical, both that such things might be true and there there's any hope of attacking such questions with current methods. | |
| Feb 28, 2017 at 17:43 | comment | added | Will Orrick | @Gerry Unfortunately not much. The writeup mentioned in my comment above is arxiv.org/abs/1112.4160. It is not published yet due mostly to inattention on my part and partly to lack of enthusiasm for this kind of work from the referees. I did have some ideas a few years back for how to adapt the method for the $11\times 11$ and possibly $13\times 13$ cases, at some increase in complexity of the code and decrease in its efficiency, but have not seriously pursued it. I believe there's been no progress related to any of the conjectures in my answer below. Experts I've spoken with... | |
| Feb 25, 2017 at 22:09 | comment | added | Gerry Myerson | @Will, any update? | |
| S Feb 25, 2017 at 15:52 | history | suggested | Rodrigo de Azevedo |
Added relevant tag
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| Feb 25, 2017 at 15:40 | review | Suggested edits | |||
| S Feb 25, 2017 at 15:52 | |||||
| Dec 31, 2016 at 14:41 | history | edited | Qfwfq | CC BY-SA 3.0 |
added 12 characters in body; edited title
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| Apr 15, 2010 at 22:25 | answer | added | Will Orrick | timeline score: 4 | |
| Apr 15, 2010 at 3:27 | comment | added | Will Orrick | The 12 x 12 case has recently been settled by a calculation of Richard Brent, Judy-anne Osborn, Paul Zimmermann, and myself: the conjecture at indiana.edu/~maxdet/spectrum13.html is correct. (The link is to the n=13 case of {-1,1} matrices, which corresponds to the 12 x 12 case of {0,1} matrices.) We hope to write up these and other results soon. The answer for 11 x 11 is still unknown, as our code works only for odd-sized {-1,1} matrices. | |
| Mar 19, 2010 at 15:52 | vote | accept | Ross Snider | ||
| Mar 18, 2010 at 18:36 | comment | added | Ross Snider | Ahh, yes there seems to be a small problem with the determinant calculating code. I grabbed a determinant function that works with libboost online. Should have checked it for accuracy in all cases. | |
| Mar 18, 2010 at 14:51 | comment | added | Gerhard Paseman | I also did the smaller cases by hand. For 4x4 I got 7, for 5x5 I got 11 and 6x6 I got 19. You might check your computer program. Gerhard "Ask Me About System Design" Paseman, 2010.03.18 | |
| Mar 18, 2010 at 7:27 | comment | added | Gerhard Paseman | I remember doing the 7x7 spectrum by hand years ago after some computer searching with a single desktop computer of 1990's vintage. It feels good to be compared to a computing cluster. Gerhard "Ask Me About System Design" Paseman, 2010.03.18 | |
| Mar 18, 2010 at 7:22 | answer | added | Gerhard Paseman | timeline score: 4 | |
| Mar 18, 2010 at 6:52 | comment | added | Cam McLeman | Just a comment that you can certainly improve the efficiency of your brute force approach. For example, once you have one determinant calculated, you may as well cross off all matrices which are row or column swaps. While we're at it, the 6x6 determinants properly include into the 7x7 determinants (multiple times!), so you could skip over these if you've already done them. And I'm sure you could be much more clever than these...this was just a first thought. | |
| Mar 18, 2010 at 6:40 | answer | added | Gerry Myerson | timeline score: 3 | |
| Mar 18, 2010 at 6:30 | comment | added | Gerry Myerson | I'm sorry, I was a bit careless above; the 10 x 10 number is 269, but the 11 x 11 is unknown. See A013588 in the Online Encyclopedia of Integer Sequences, oeis.org/A013588 | |
| Mar 18, 2010 at 6:27 | comment | added | Gerry Myerson | I think this is a hard problem. As evidence, I note that the smallest positive integer which is not the determinant of a 9 x 9 binary matrix was only recently found to be 103 (arXiv.org/pdf/math/0511636v1); the smallest positive integer not the determinant of a 10 x 10 binary matrix seems to be unknown. | |
| Mar 18, 2010 at 6:26 | comment | added | Qiaochu Yuan | Singular matrices certainly do have a determinant; it's zero... | |
| Mar 18, 2010 at 6:02 | history | asked | Ross Snider | CC BY-SA 2.5 |