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

Baoulina, Ioulia N.

doi:10.5802/jtnb.747

Baoulina, Ioulia N.¹

¹ Statistics and Mathematics Unit Indian Statistical Institute 8th Mile, Mysore Road R. V. College Post Bangalore 560059, India

Journal de théorie des nombres de Bordeaux, Volume 23 (2011) no. 1, pp. 1-20.

Abstract
Résumé

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.

On considère une équation de la forme suivante

a_{1}^{} x_{1}^{2} + \dots + a_{n}^{} x_{n}^{2} = b x_{1} \dots x_{n}

sur le corps fini $𝔽_{q} = 𝔽_{p^{s}}$ . Carlitz a obtenu des formules pour le nombre de solutions de cette équation dans le cas $n = 3$ et le cas $n = 4$ avec $q \equiv 3 (mod 4)$ . Dans des travaux anciens, on a démontré des formules pour le nombre de solutions lorsque $d = gcd (n - 2, (q - 1) / 2) = 1$ ou $2$ ou $4$ , et aussi lorsque $d > 1$ et $- 1$ est une puissance de $p$ modulo $2 d$ . Dans ce papier, on démontre des formules pour le nombre de solutions lorsque $d = 2^{t}$ , $t \geq 3$ , $p \equiv 3 ou 5 (mod 8)$ ou $p \equiv 9 (mod 16)$ . On obtient aussi une borne inférieure pour le nombre de solutions dans le cas général.

MR Zbl

DOI: 10.5802/jtnb.747

Author's affiliations:

Baoulina, Ioulia N. ¹

¹ Statistics and Mathematics Unit Indian Statistical Institute 8th Mile, Mysore Road R. V. College Post Bangalore 560059, India

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     title = {On the {Carlitz} problem on the number of solutions to some special equations over finite fields},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
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Baoulina, Ioulia N. On the Carlitz problem on the number of solutions to some special equations over finite fields. Journal de théorie des nombres de Bordeaux, Volume 23 (2011) no. 1, pp. 1-20. doi : 10.5802/jtnb.747. http://www.numdam.org/articles/10.5802/jtnb.747/

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