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

Ioulia N. Baoulina

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

Ioulia N. Baoulina^[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 (2011)

Volume: 23, Issue: 1, page 1-20
ISSN: 1246-7405

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Abstract

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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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RIS

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 23.1 (2011): 1-20. <http://eudml.org/doc/219782>.

@article{Baoulina2011,
abstract = {We consider an equation of the type\[ a\_1^\{\}x\_1^2+\dots +a\_n^\{\}x\_n^2=bx\_1\cdots x\_n \]over the finite field $\mathbb\{F\}_q=\mathbb\{F\}_\{p^s\}$. Carlitz obtained formulas for the number of solutions to this equation when $n=3$ and when $n=4$ and $q\equiv 3\hspace\{4.44443pt\}(\@mod \; 4)$. 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 $2d$. In this paper, we obtain formulas for the number of solutions when $d=2^t$, $t\ge 3$, $p\equiv 3\,\,\text\{or $5\!\hspace\{4.44443pt\}(\@mod \; 8)$\}$ or $p\equiv 9\!\hspace\{4.44443pt\}(\@mod \; 16)$. For general case, we derive lower bounds for the number of solutions.},
affiliation = {Statistics and Mathematics Unit Indian Statistical Institute 8th Mile, Mysore Road R. V. College Post Bangalore 560059, India},
author = {Baoulina, Ioulia N.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {finite fields; Gauss sums},
language = {eng},
month = {3},
number = {1},
pages = {1-20},
publisher = {Société Arithmétique de Bordeaux},
title = {On the Carlitz problem on the number of solutions to some special equations over finite fields},
url = {http://eudml.org/doc/219782},
volume = {23},
year = {2011},
}

TY - JOUR
AU - Baoulina, Ioulia N.
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
DA - 2011/3//
PB - Société Arithmétique de Bordeaux
VL - 23
IS - 1
SP - 1
EP - 20
AB - We consider an equation of the type\[ a_1^{}x_1^2+\dots +a_n^{}x_n^2=bx_1\cdots x_n \]over the finite field $\mathbb{F}_q=\mathbb{F}_{p^s}$. Carlitz obtained formulas for the number of solutions to this equation when $n=3$ and when $n=4$ and $q\equiv 3\hspace{4.44443pt}(\@mod \; 4)$. 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 $2d$. In this paper, we obtain formulas for the number of solutions when $d=2^t$, $t\ge 3$, $p\equiv 3\,\,\text{or $5\!\hspace{4.44443pt}(\@mod \; 8)$}$ or $p\equiv 9\!\hspace{4.44443pt}(\@mod \; 16)$. For general case, we derive lower bounds for the number of solutions.
LA - eng
KW - finite fields; Gauss sums
UR - http://eudml.org/doc/219782
ER -

References

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I. Baoulina, On the problem of explicit evaluation of the number of solutions of the equation $a_{1}^{} x_{1}^{2} + \dots + a_{n}^{} x_{n}^{2} = b x_{1}^{} \dots x_{n}^{}$ 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. Zbl1086.11021 MR1925639
I. Baoulina, On some equations over finite fields. J. Théor. Nombres Bordeaux 17 (2005), 45–50. Zbl1119.11033 MR2152209
I. Baoulina, Generalizations of the Markoff-Hurwitz equations over finite fields. J. Number Theory 118 (2006), 31–52. Zbl1094.11024 MR2220260
I. Baoulina, On the number of solutions to the equation ${(x_{1} + \dots + x_{n})}^{2} = a x_{1} \dots x_{n}$ in a finite field. Int. J. Number Theory 4 (2008), 797–817. Zbl1204.11067 MR2458844
A. Baragar, The Markoff Equation and Equations of Hurwitz. Ph. D. Thesis, Brown University, 1991. MR2686830
B. C. Berndt, R. J. Evans and K. S. Williams, Gauss and Jacobi Sums. Wiley-Interscience, New York, 1998. Zbl0906.11001 MR1625181
L. Carlitz, Certain special equations in a finite field. Monatsh. Math. 58 (1954), 5–12. Zbl0055.26803 MR61121
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. Zbl0602.12005 MR908829
R. Lidl and H. Niederreiter, Finite Fields. Cambridge Univ. Press, Cambridge, 1997. Zbl0866.11069 MR1429394

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