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.

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.

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.

Published online:
DOI: 10.5802/jtnb.747
Ioulia N. Baoulina 1

1 Statistics and Mathematics Unit Indian Statistical Institute 8th Mile, Mysore Road R. V. College Post Bangalore 560059, India
@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/}
}
TY  - JOUR
TI  - On the Carlitz problem on the number of solutions to some special equations over finite fields
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2011
DA  - 2011///
SP  - 1
EP  - 20
VL  - 23
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.747/
UR  - https://zbmath.org/?q=an%3A1267.11035
UR  - https://www.ams.org/mathscinet-getitem?mr=2780616
UR  - https://doi.org/10.5802/jtnb.747
DO  - 10.5802/jtnb.747
LA  - en
ID  - JTNB_2011__23_1_1_0
ER  - 
%0 Journal Article
%T On the Carlitz problem on the number of solutions to some special equations over finite fields
%J Journal de Théorie des Nombres de Bordeaux
%D 2011
%P 1-20
%V 23
%N 1
%I Société Arithmétique de Bordeaux
%U https://doi.org/10.5802/jtnb.747
%R 10.5802/jtnb.747
%G en
%F JTNB_2011__23_1_1_0
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, Volume 23 (2011) no. 1, pp. 1-20. doi : 10.5802/jtnb.747. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.747/

[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. | MR: 1925639 | Zbl: 1086.11021

[2] I. Baoulina, On some equations over finite fields. J. Théor. Nombres Bordeaux 17 (2005), 45–50. | Numdam | MR: 2152209 | Zbl: 1119.11033

[3] I. Baoulina, Generalizations of the Markoff-Hurwitz equations over finite fields. J. Number Theory 118 (2006), 31–52. | MR: 2220260 | Zbl: 1094.11024

[4] I. Baoulina, On the number of solutions to the equation (x 1 ++x n ) 2 =ax 1 x n in a finite field. Int. J. Number Theory 4 (2008), 797–817. | MR: 2458844 | Zbl: pre05499308

[5] A. Baragar, The Markoff Equation and Equations of Hurwitz. Ph. D. Thesis, Brown University, 1991. | MR: 2686830

[6] B. C. Berndt, R. J. Evans and K. S. Williams, Gauss and Jacobi Sums. Wiley-Interscience, New York, 1998. | MR: 1625181 | Zbl: 0906.11001

[7] L. Carlitz, Certain special equations in a finite field. Monatsh. Math. 58 (1954), 5–12. | MR: 61121 | Zbl: 0055.26803

[8] S. A. Katre and A. R. Rajwade, Resolution of the sign ambiguity in the determination of the cyclotomic numbers of order 4 and the corresponding Jacobsthal sum. Math. Scand. 60 (1987), 52–62. | MR: 908829 | Zbl: 0602.12005

[9] R. Lidl and H. Niederreiter, Finite Fields. Cambridge Univ. Press, Cambridge, 1997. | MR: 1429394 | Zbl: 0866.11069

Cited by Sources: