The local sum conjecture in two dimensions.

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. [ABSTRACT FROM AUTHOR]