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

Keywords:

equations over finite fields

Zbl 0055.26803