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Factorization formulae on counting zeros of diagonal equations over finite fields

Authors:	Wei Cao Qi Sun

Institution:	Mathematical College, Sichuan University, Chengdu 610064, People's Republic of China ; Mathematical College, Sichuan University, Chengdu 610064, People's Republic of China

Abstract:	Let be the number of solutions $(u_1,\ldots,u_n)$ of the equation $a_1u_1^{d_1}+\cdots+a_nu_n^{d_n}=0$ over the finite field , and let be the number of solutions of the equation $\sum_{i=1}^nx_i/d_i\equiv 0\pmod{1}, 1\leqslant x_i\leqslant d_i-1$ . If , let be the least integer represented by $\sum_{i=1}^nx_i/d_i, 1\leqslant x_i\leqslant d_i-1$ . and play important roles in estimating . Based on a partition of $\{d_1,\dots,d_n\}$ , we obtain the factorizations of and , respectively. All these factorizations can simplify the corresponding calculations in most cases or give the explicit formulae for in some special cases.

Keywords:	Jacobi sum Gauss sum diagonal equation finite fields

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