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arXiv · 1107.3367

The probability of rectangular unimodular matrices over $\F_q[x]$

Xiangqian Guo·Guangyu Yang
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Abstract

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.

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Xiangqian Guo, Guangyu Yang. 2013-01-16. The probability of rectangular unimodular matrices over $\F_q[x]$. https://arxiv.org/abs/1107.3367

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