Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T04:13:47.851Z Has data issue: false hasContentIssue false
  • English
  • Français

Short Kloosterman Sums for Polynomials over Finite Fields

Published online by Cambridge University Press:  20 November 2018

William D. Banks ,
Asma Harcharras  and
Igor E. Shparlinski
William D. Banks
Affiliation:
Department of Mathematics, University of Missouri Columbia, Missouri, 65211, USA, e-mail: [email protected]
Asma Harcharras
Affiliation:
Department of Mathematics, University of Missouri Columbia, Missouri, 65211, USA, e-mail: [email protected]
Igor E. Shparlinski
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We extend to the setting of polynomials over a finite field certain estimates for short Kloosterman sums originally due to Karatsuba. Our estimates are then used to establish some uniformity of distribution results in the ring ${{\mathbb{F}}_{q}}\left[ x \right]\,/\,M\left( x \right)$ for collections of polynomials either of the form ${{f}^{-1}}{{g}^{-1}}$ or of the form ${{f}^{-1}}{{g}^{-1}}\,+\,afg$, where $f$ and $g$ are polynomials coprime to $M$ and of very small degree relative to $M$, and $a$ is an arbitrary polynomial. We also give estimates for short Kloosterman sums where the summation runs over products of two irreducible polynomials of small degree. It is likely that this result can be used to give an improvement of the Brun-Titchmarsh theorem for polynomials over finite fields.

Keywords

11T2311T06
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 55 , Issue 2 , 01 April 2003 , pp. 225 - 246
Copyright
Copyright © Canadian Mathematical Society 2003

References

[1] Banks, W. and Shparlinski, I. E., Distribution of inverses in polynomial rings. Indag.Math. 12 (2001), 303–315.Google Scholar
[2] Friedlander, J. and Iwaniec, H., The Brun-Titchmarsh theorem. Analytic Number Theory, Lond. Math. Soc. Lecture Note Series 247 (1997), 363–372.Google Scholar
[3] Hsu, C.-N., The Brun-Titchmarsh theorem in function fields. J. Number Theory 79 (1999), 67–82.Google Scholar
[4] Karatsuba, A. A., Fractional parts of functions of a special form. Izv. Akad. Nauk Ser. Mat., Transl. as Russian Acad. Sci. Izv.Math. (4) 55 (1995), 61–80.Google Scholar
[5] Karatsuba, A. A., Analogues of Kloosterman sums. (Russian), Izv. Ross. Akad. Nauk Ser. Mat. (5) 59 (1995), 93–102.Google Scholar