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Fix numbers $m, n, k\in {\Bbb Z}_+$ and $r\in {\Bbb R}_+$.

What non-trivial estimates exist for the probability that a random $m\times n$ matrix, with integer entries and with all its rows of Euclidean norm less than or equal to $r$, will have rank $k$?

I'm particularly interested in results asymptotic in the variable $r$. (I've worked out the coefficient for $m=n=2, k=1$, but my method doesn't seem to generalize.)

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If you get no other data, look up Miodrag Zivkovic and classification of 0-1 matrices. That and one other work (I think of his, but my memory is not fully cooperating right now) consider ranks for 0-1 matrices, and may give you something that you can use. It will not directly address your question though. Gerhard "Ask Me About System Design" Paseman, 2012.04.25 – Gerhard Paseman Apr 25 '12 at 18:08
up vote 5 down vote accepted

This is addressed in

MR1169034 (94e:11073) Katznelson, Yonatan R.(1-MSRI) Integral matrices of fixed rank. Proc. Amer. Math. Soc. 120 (1994), no. 3, 667–675.

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Thank you Igor for this reference! I see that Katznelson actually treats a very closely related problem - he takes a disk in ${\Bbb R}^{m\times n}$ where I asked about a polydisk. I haven't had enough time with the paper yet to judge whether that makes much difference. – David Feldman Apr 25 '12 at 22:42
It does not make any difference, the percentage of the matrices you want is the same, as long as the domain is reasonably regular (which the polydisk certainly is). – Igor Rivin Apr 25 '12 at 22:48

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