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

Baoulina, Ioulia

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].

Reviewer: L. N. Vaserstein (University Park)

MSC:

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

Keywords:

equations over finite fields

Citations:

Zbl 0055.26803