Let $ \mathbb{F}_q $ be a finite field of $ q $ elements. For $ n\in\mathbb{N}^* $ with $ n\ge2, $ let $ M_{n}: = Mat_n(\mathbb{F}_q) $ be the ring of matrices of order $ n $ over $ \mathbb{F}_q, $ $ G_{n, 1}: = Sl_n(\mathbb{F}_q) $ be the special linear group over $ \mathbb{F}_q. $ In this paper, by using the technique of Fourier transformation, we obtain a formula for the number of representations of any element of $ M_{n} $ as the sum of $ k $ matrices in $ G_{n, 1}. $ As a corollary, we give another proof of the number of the third power moment of the classic Kloosterman sum.
Citation: Yifan Luo, Qingzhong Ji. On the sum of matrices of special linear group over finite field[J]. AIMS Mathematics, 2025, 10(2): 3642-3651. doi: 10.3934/math.2025168
Let $ \mathbb{F}_q $ be a finite field of $ q $ elements. For $ n\in\mathbb{N}^* $ with $ n\ge2, $ let $ M_{n}: = Mat_n(\mathbb{F}_q) $ be the ring of matrices of order $ n $ over $ \mathbb{F}_q, $ $ G_{n, 1}: = Sl_n(\mathbb{F}_q) $ be the special linear group over $ \mathbb{F}_q. $ In this paper, by using the technique of Fourier transformation, we obtain a formula for the number of representations of any element of $ M_{n} $ as the sum of $ k $ matrices in $ G_{n, 1}. $ As a corollary, we give another proof of the number of the third power moment of the classic Kloosterman sum.
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Yifan Luo, Qingzhong Ji. On the sum of matrices of special linear group over finite field[J]. AIMS Mathematics, 2025, 10(2): 3642-3651. doi: 10.3934/math.2025168
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