Counting zero kernel pairs over a finite field

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• 作者：Samrith Ram ^{samrith@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：93B05 ; 93B07 ; 15B33 ; 15A22 ; 15A83
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：15 April 2016
• 年：2016
• 卷：495
• 期：Complete
• 页码：1-10
• 全文大小：267 K

文摘

Helmke et al. have recently given a formula for the number of reachable pairs of matrices over a finite field. We give a new and elementary proof of the same formula by solving the equivalent problem of determining the number of so called zero kernel pairs over a finite field. We show that the problem is equivalent to certain other enumeration problems and outline a connection with some recent results of Guo and Yang on the natural density of rectangular unimodular matrices over F_q[x]${\mathbb{F}}_q[x]$. We also propose a new conjecture on the density of unimodular matrix polynomials.