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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: bbanks@math.missouri.edu
Asma Harcharras
Affiliation:
Department of Mathematics, University of Missouri Columbia, Missouri, 65211, USA, e-mail: harchars@math.missouri.edu
Igor E. Shparlinski
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia, e-mail: igor@ics.mq.edu.au
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Abstract

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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

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