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

Abstract

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We consider an equation of the type a 1 x 1 2 + ⋯ + a n x n 2 = b x 1 ⋯ 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 ≡ 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 ≥ 3 , p ≡ 3 or 5 ( mod 8 ) or p ≡ 9 ( mod 16 ) . For general case, we derive lower bounds for the number of solutions.

How to cite

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

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