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Counting Singular Matrices with Primitive Row Vectors

Igor Wigman
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Abstract

We solve an asymptotic problem in the geometry of numbers, where we count the number of singular $n\times n$ matrices where row vectors are primitive and of length at most T. Without the constraint of primitivity, the problem was solved by Y. Katznelson. We show that as $T \to \infty $, the number is asymptotic to $ \frac{(n-1)u_n}{ζ(n) ζ(n-1)^{n}}T^{n^{2}-n}\log (T)$ for $n \ge 3$. The 3-dimensional case is the most problematic and we need to invoke an equidistribution theorem due to W. Schmidt.

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Igor Wigman. 2003-11-30. Counting Singular Matrices with Primitive Row Vectors. https://arxiv.org/abs/math/0305066

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