Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The probability of rectangular unimodular matrices over $\F_q[x]$   In this note, we compute the probability that a $k\times n$ matrix can be
extended to an $n\times n$ invertible matrix over $\F_q[x]$, which turns out to
be $(1-q^{k-n})(1-q^{k-1-n})...(1-q^{1-n})$. Connections with Dirichlet's
density theorem on the co-prime integers and its various generalizations are
also presented.

26
The categories of this paper are: bayes-an
31
The categories of this paper are: ao-sci
4
The categories of this paper are: cond-mat
35
The categories of this paper are: mtrl-th
22
The categories of this paper are: chao-dyn
17
The categories of this paper are: q-fin
1
The categories of this paper are: cs
6
The categories of this paper are: astro-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: funct-an
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The categories of this paper are: solv-int
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The categories of this paper are: plasm-ph
32
The categories of this paper are: atom-ph
11
The categories of this paper are: hep-lat
28
The categories of this paper are: alg-geom
37
The categories of this paper are: supr-con