Short Kloosterman Sums for Polynomials over Finite Fields

[摘要] We extend to the setting of polynomials over a finite field certainestimates for short Kloosterman sums originally due to Karatsuba.Our estimates are then used to establish some uniformity ofdistribution results in the ring $mathbb{F}_q[x]/M(x)$ for collections ofpolynomials 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 anarbitrary polynomial. We also give estimates for short Kloostermansums where the summation runs over products of two irreduciblepolynomials of small degree. It is likely that this result can beused to give an improvement of the Brun-Titchmarsh theorem forpolynomials over finite fields.

[学科分类] 数学（综合）