The local sum conjecture is a variant of some of Igusa’s questions on exponential sums put forward by Denef and Sperber. In a remarkable paper by Cluckers, Mustata and Nguyen, this conjecture has been established in all dimensions, using sophisticated, powerful techniques from a research area blending algebraic geometry with ideas from logic. The purpose of this paper is to give an elementary proof of this conjecture in two dimensions which follows Varčenko’s treatment of Euclidean oscillatory integrals based on Newton polyhedra for good coordinate choices. Another elementary proof is given by Veys from an algebraic geometric perspective.

中文翻译：

---

二维局部和猜想

局部和猜想是由 Denef 和 Sperber 提出的 Igusa 关于指数和的一些问题的变体。在 Cluckers、Musta 和 Nguyen 的一篇出色的论文中，这个猜想已经在各个维度上得到了证实，使用了来自研究领域的复杂、强大的技术，将代数几何与逻辑思想相结合。本文的目的是在二维方面对这个猜想给出一个基本的证明，它遵循 Varčenko 对基于牛顿多面体的欧几里得振荡积分的处理，以获得良好的坐标选择。Veys 从代数几何的角度给出了另一个基本证明。