Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.)

share|improve this question
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
add comment

1 Answer

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.

share|improve this answer
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
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.