Exponential sums and polynomial congruences in two variables: the quasi-homogeneous case

Abstract

We adapt ideas of Phong, Stein and Sturm and ideas of Ikromov and Müller from the continuous setting to various discrete settings, obtaining sharp bounds for exponential sums and the number of solutions to polynomial congruences for general quasi-homogeneous polynomials in two variables. This extends work of Denef and Sperber and also Cluckers regarding a conjecture of Igusa in the two dimensional setting by no longer requiring the polynomial to be nondegenerate with respect to its Newton diagram.

Cited by in corpus (4)

• Igusa's conjecture for exponential sums: optimal estimates for non-rational singularities
• Counting factorisations of monomials over rings of integers modulo $N$
• Proof of Cluckers-Veys's conjecture on exponential sums for polynomials with log-canonical threshold at most a half
• Bounds for p-adic exponential sums and log-canonical thresholds