ON THE LOCAL SUM CONJECTURE IN TWO DIMENSIONS

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Abstract

The local sum conjecture is a variant of some of Igusa’s questions
on exponential sums put forward by Denef and Sperber in [7]. In a remarkable
paper [6] 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 Varˇcenko’s treatment of euclidean oscillatory integrals based on
Newton polyhedra for good coordinate choices. Another elementary proof is
given by Veys [18] from an algebraic geometric perspective.
Original languageEnglish
Number of pages32
JournalInternational Journal of Number Theory
Publication statusAccepted/In press - 26 Feb 2020

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