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.
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机译：我们扩展到有限域确定性上的多项式的设置，该确定性最初是由Karatsuba导致的短Kloosterman和的。我们的估计然后用于建立环$ mathbb {F} _q [x] / M（x）$的分布结果的均匀性用于$ f ^ {-1} g ^ {-1} $形式或$ f ^ {-1} g ^ {-1} + afg $形式的多项式集合，其中$ f $和$ g $是$ M $的多项式的互质数，相对于$ M $的程度很小，而$ a $是任意多项式。我们还给出了短Kloostermansum的估计，其中总和超过了两个不可约的小次数多项式的乘积。该结果很可能可以用来改进有限域上的Brun-Titchmarsh定理多项式。
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