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

Citations:

Zbl 0055.26803; Zbl 1086.11021
Full Text:

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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