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On the problem of explicit evaluation of the number of solutions of the equation \(a_1x_1^2+\cdots+ a_nx_n^2= bx_1\cdots x_n\) in a finite field. (English) Zbl 1086.11021

Adhikari, S.D. (ed.) et al., Current trends in number theory. Proceedings of the international conference on number theory, Allahabad, India, November 2000. New Delhi: Hindustan Book Agency (ISBN 81-85931-33-X/hbk). 27-37 (2002).
The number \(N_q\) of all solutions in a finite field \(\mathbb F_q\) for the equation \[ a_1x_1^2+\cdots+a_nx_n^2=bx_1 \cdots x_n \] with given odd \(q\) and nonzero \(b,a_i \in \mathbb F_q,\) is evaluated. The cases \(n=3, 4\) were considered previously by L. Carlitz [Monatsh. Math. 58, 5–12 (1954; Zbl 0055.26803)].
For the entire collection see [Zbl 1003.00014].

MSC:

11D79 Congruences in many variables
11G25 Varieties over finite and local fields
11T24 Other character sums and Gauss sums

Citations:

Zbl 0055.26803
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