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28 March 2024
 
  » arxiv » 0903.2508

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Distribution of determinant of matrices with restricted entries over finite fields
Le Anh Vinh ;
Date 13 Mar 2009
AbstractFor a prime power $q$, we study the distribution of determinent of matrices with restricted entries over a finite field $mathbbm{F}_q$ of $q$ elements. More precisely, let $N_d (mathcal{A}; t)$ be the number of $d imes d$ matrices with entries in $mathcal{A}$ having determinant $t$. We show that [ N_d (mathcal{A}; t) = (1 + o (1)) frac{|mathcal{A}|^{d^2}}{q}, ] if $|mathcal{A}| = omega(q^{frac{d}{2d-1}})$, $dgeqslant 4$. When $q$ is a prime and $mathcal{A}$ is a symmetric interval $[-H,H]$, we get the same result for $dgeqslant 3$. This improves a result of Ahmadi and Shparlinski (2007).
Source arXiv, 0903.2508
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