The number of solutions to the alternate matrix equation over a finite field and a q-identity |
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| Institution: | 1. Department of Mathematics, Penn State University, PA, United States;2. Department of Computer Science, Rutgers, The State University of New Jersey, NJ, United States;3. School of Mathematics and Statistics, Wuhan University, China;1. School of Mathematical and Statistical Sciences, North-West University, Research Focus: Pure and Applied Analytics, Private Bag X6001, Potchefstroom 2520, South Africa;2. Department of Mathematics, Faculty of Science, VU Amsterdam, De Boelelaan 1111, 1081 HV Amsterdam, the Netherlands;3. Research Focus: Pure and Applied Analytics, North-West University, Potchefstroom, South Africa;4. DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa |
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| Abstract: | Let Fq be a finite field with q elements, where q is a power of a prime. In this paper, we first correct a counting error for the formula N(K2ν,0(m)) occurring in Carlitz (1954. Arch. Math. V, 19–31). Next, using the geometry of symplectic group over Fq, we have given the numbers of solutions X of rank k and solutions X to equation XAX′=B over Fq, where A and B are alternate matrices of order n, rank 2ν and order m, rank 2s, respectively. Finally, an elementary q-identity is obtained from N(K2ν,0(0)), and the explicit results for N(Kn,2ν,Km,2s) is represented by terminating q-hypergeometric series. |
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