On some equations over finite fields. (English) Zbl 1119.11033

Let \(\mathbb F_q\) be a finite field with \(q=p^s\) elements and let \(N_q\) be the number of solutions in \(\mathbb F_q\) of \(\sum_{i=1}^n a_ix_i=bx_1\cdots x_n\). L. Carlitz [Monatsh. Math. 58, 5–12 (1954; Zbl 0055.26803)] gave formulas for \(N_q\) when \(n=3\) and when \(n=4\), \(q\equiv 3\pmod{4}\). In a previous paper [Current Trends in Number Theory. Allahabad, India, November 2000. New Delhi: Hindustan Book Agency, 27–37 (2002; Zbl 1086.11021)], the author found explicit formulas for \(N_q\) when \(d=(n-2, (q-1)/2)\) is \(1\) or \(2\). Here formulas are found when \(d\) divides some \(p^{\ell}+1\) and certain parity conditions hold. The results are obtained by determining certain Gauss sums.


11G25 Varieties over finite and local fields
11T24 Other character sums and Gauss sums
11D72 Diophantine equations in many variables
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[1] I. Baoulina, On the Problem of Explicit Evaluation of the Number of Solutions of the Equation \(a_1^{}x_1^2+\cdots +a_n^{}x_n^2=bx_1^{}\cdots x_n^{}\) in a Finite Field. In Current Trends in Number Theory, Edited by S. D. Adhikari, S. A. Katre and B. Ramakrishnan, Hindustan Book Agency, New Delhi, 2002, 27-37. · Zbl 1086.11021
[2] A. Baragar, The Markoff Equation and Equations of Hurwitz. Ph. D. Thesis, Brown University, 1991.
[3] B. C. Berndt, R. J. Evans, K. S. Williams, Gauss and Jacobi Sums. Wiley-Interscience, New York, 1998. · Zbl 0906.11001
[4] L. Carlitz, Certain special equations in a finite field. Monatsh. Math. 58 (1954), 5-12. · Zbl 0055.26803
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