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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Zeros of diagonal equations over finite fields


Author: Da Qing Wan
Journal: Proc. Amer. Math. Soc. 103 (1988), 1049-1052
MSC: Primary 11T41
MathSciNet review: 954981
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Abstract: Let $ N$ be the number of solutions $ ({x_1}, \ldots ,{x_n})$ of the equation (1)

$\displaystyle (1)\quad {c_1}x_1^{{d_1}} + {c_2}x_2^{{d_2}} + \cdots + {c_n}x_n^{{d_n}} = c$

over the finite field $ {F_q}$, where $ {d_i}\vert(q - 1),{c_i} \in F_q^*(i = 1, \ldots ,n)$, and $ c \in {F_q}$. If

$\displaystyle \frac{1} {{{d_1}}} + \frac{1} {{{d_2}}} + \cdots + \frac{1} {{{d_n}}} > b \geq 1$

for some positive integer $ b$, we prove that $ {q^b}\vert N$. This result is an improvement of the theorem that $ p\vert N$ obtained by B. Morlaye [7] and also by J. R. Joly [3].

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0954981-2
PII: S 0002-9939(1988)0954981-2
Article copyright: © Copyright 1988 American Mathematical Society