发表机构
Georg-August-Universität Göttingen（哥廷根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对素数p，在非齐次多项式的短Dirichlet特征和问题上，开发新Burgess放大方法突破原Burgess界，在k≥6时取得短盒子和的非平凡界，为非齐次情形下Burgess型方法的首次成功应用。
AI 中文摘要
设p为素数，我们证明了模p的Dirichlet特征在一类多项式（n个变量、次数为k，不一定是齐次的）上的短和的非平凡界。对于大的n，我们进一步在边长短至p^{1/(k-1)+ε}的盒子上的和取得非平凡界，当k≥6时，这突破了Burgess的p^{1/4+ε}障碍。证明中，我们开发了Burgess放大方法的新变体，将问题简化为估计加法特征和，这是Burgess型方法在非齐次情形下首次成功的案例。
英文摘要
Let $p$ be a prime. We prove nontrivial bounds on short sums of Dirichlet characters mod $p$ evaluated at a class of polynomials, not necessarily homogeneous, in $n$ variables and of degree $k$. For large $n$, we further achieve nontrivial bounds for sums over boxes with side-lengths as short as $p^{1/(k-1)+\varepsilon}$, which breaks past the Burgess barrier of $p^{1/4+\varepsilon}$ as soon as $k\geq 6$. In the proof, we develop a new variation of the Burgess amplification method that reduces the problem to bounding additive character sums. This is the first case in which a Burgess-type method succeeds in the inhomogeneous setting.
Comments25 pages
