New bounds for exponential sums with a non-degenerate phase polynomial

TL;DR

This paper establishes new bounds for exponential sums with non-degenerate phase polynomials, confirming conjectures related to Igusa's local zeta functions and improving understanding of exponential sum behavior modulo prime powers.

Contribution

It proves a conjecture on exponential sums modulo p^m for non-degenerate polynomials and provides an improved bound on related sums over finite fields.

Findings

Proved conjecture for exponential sums with non-degenerate phase polynomials.

Established an improved bound on exponential sums over finite fields.

Settled a conjecture on the leading Taylor coefficient of Igusa's local zeta function.

Abstract

We prove a recent conjecture due to Cluckers and Veys on exponential sums modulo $p^m$ for $m \geq 2$ in the special case where the phase polynomial $f$ is sufficiently non-degenerate with respect to its Newton polyhedron at the origin. Our main auxiliary result is an improved bound on certain related exponential sums over finite fields. This bound can also be used to settle a conjecture of Denef and Hoornaert on the candidate-leading Taylor coefficient of Igusa's local zeta function associated to a non-degenerate polynomial, at its largest non-trivial real candidate pole.