In recent years, there has been a lot of progress in obtaining non-trivial bounds for bilinear forms of Kloosterman sums in $\mathbb{Z}/m\mathbb{Z}$ for arbitrary integers m. These results have been motivated by a wide variety of applications, such as improved asymptotic formulas for moments of L-functions. However, there has been very little work done in this area in the setting of rational function fields over finite fields. We remedy this and provide a number of new non-trivial bounds for bilinear forms of Kloosterman and Gauss sums in this setting, based on new bounds on the number of solutions to certain modular congruences in ${\mathbb{F}}_q[T]$. These improve upon some results of Macourt and Shparlinski (2019).

近年来，在获得双线性形式 Kloosterman 和的非平凡界限方面取得了很大进展。$\mathbb{Z}/米\mathbb{Z}$对于任意整数m。这些结果受到了广泛应用的推动，例如L函数矩的改进渐近公式。然而，在有限域上有理函数域的设置方面，这方面的工作还很少。我们对此进行了补救，并基于对某些模同余的解的数量的新界限，在此设置中为 Kloosterman 和高斯和的双线性形式提供了许多新的非平凡界限。${\mathbb{F}}_q[\mathrm{时间}]$。这些改进了 Macourt 和 Shparlinski (2019) 的一些结果。