Research on a Class of Equations over Finite Fields |
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Authors: | Shuangnian Hu Tianbo Diao Yujun Niu Honge Wu |
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Affiliation: | 1.School of Mathematics and Statistics,Nanyang Institute of Technology,Nanyang, Henan,China |
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Abstract: | Let Fq stand for the finite field of odd characteristic p with q elements ( q = pn,n∈N ) and Fq* denote the set of all the nonzero elements of Fq. In this paper, by using the augmented degree matrix and the result given by Cao, we obtain a formula for the number of rational points of the following equation over ({F_q}:f({x_1},{x_2},...{x_n}) = {({a_1}{x_1}{x_2} + {a_1}{x_2}x_3^d + {a_{n - 1}}{x_{n - 1}}x_1^d + {a_n}{x_n}x_1^d)^lambda } - bx_1^{{d_1}}x_n^{{d_n}}...x_n^{{d_n}},with{a_i},b in F_q^*,n geqslant 2,lambda > 0) being positive integers, and d, di being nonnegative integers for 1 in. This technique can be applied to the polynomials of the form h1λ=h2 with λ being positive integer and ({h_1},{h_2} in {F_q}[{x_1},{x_2},...{x_n}]). It extends the results of the Markoff- Hurwitz-type equations. |
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