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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.

MSC:

11G25 Varieties over finite and local fields
11T24 Other character sums and Gauss sums
11D72 Diophantine equations in many variables
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References:

[1] I. Baoulina, On the Problem of Explicit Evaluation of the Number of Solutions of the Equation \(a_1^{}x_1^2+⋯ +a_n^{}x_n^2=bx_1^{}⋯ x_n^{}\) 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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