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On the Carlitz problem on the number of solutions to some special equations over finite fields

Ioulia N. Baoulina — 2011

Journal de Théorie des Nombres de Bordeaux

We consider an equation of the type $a_{1}^{} x_{1}^{2} + \dots + a_{n}^{} x_{n}^{2} = b x_{1} \dots x_{n}$ over the finite field $𝔽_{q} = 𝔽_{p^{s}}$ . Carlitz obtained formulas for the number of solutions to this equation when $n = 3$ and when $n = 4$ and $q \equiv 3 (mod 4)$ . In our earlier papers, we found formulas for the number of solutions when $d = gcd (n - 2, (q - 1) / 2) = 1$ or $2$ or $4$ ; and when $d > 1$ and $- 1$ is a power of $p$ modulo $2 d$ . In this paper, we obtain formulas for the number of solutions when $d = 2^{t}$ , $t \geq 3$ , $p \equiv 3 or 5 (mod 8)$ or $p \equiv 9 (mod 16)$ . For general case, we derive lower bounds for the number of solutions.

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