On the Carlitz problem on the number of solutions to some special equations over finite fields
Journal de Théorie des Nombres de Bordeaux, Tome 23 (2011) no. 1, pp. 1-20.

On considère une équation de la forme suivante

a1x12++anxn2=bx1xn

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 q3(mod4). 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 2d. Dans ce papier, on démontre des formules pour le nombre de solutions lorsque d=2 t , t3, p3ou5(mod8) ou p9(mod16). On obtient aussi une borne inférieure pour le nombre de solutions dans le cas général.

We consider an equation of the type

a1x12++anxn2=bx1xn

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 q3(mod4). 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 2d. In this paper, we obtain formulas for the number of solutions when d=2 t , t3, p3or5(mod8) or p9(mod16). For general case, we derive lower bounds for the number of solutions.

@article{JTNB_2011__23_1_1_0,
     author = {Ioulia N. Baoulina},
     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},
     pages = {1--20},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {23},
     number = {1},
     year = {2011},
     doi = {10.5802/jtnb.747},
     zbl = {1267.11035},
     mrnumber = {2780616},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.747/}
}
Ioulia N. Baoulina. On the Carlitz problem on the number of solutions to some special equations over finite fields. Journal de Théorie des Nombres de Bordeaux, Tome 23 (2011) no. 1, pp. 1-20. doi : 10.5802/jtnb.747. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.747/

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