Igusa's conjecture for exponential sums: optimal estimates for non-rational singularities

TL;DR

This paper establishes optimal bounds for exponential sums related to hypersurfaces with non-rational singularities by connecting log canonical thresholds to motivic oscillation indices, advancing Igusa's conjecture.

Contribution

It introduces a new method linking log canonical thresholds with motivic oscillation indices to prove generalized bounds in Igusa's conjecture for non-rational singularities.

Findings

Proved an upper bound on the log canonical threshold for certain hypersurfaces.

Generalized Igusa's conjecture to all non-rational singularities.

Demonstrated optimality of estimates through threshold comparisons.

Abstract

We prove an upper bound on the log canonical threshold of a hypersurface that satisfies a certain power condition and use it to prove several generalizations of Igusa's conjecture on exponential sums, with the log-canonical threshold in the exponent of the estimates. We show that this covers optimally all situations of the conjectures for non-rational singularities, by comparing the log canonical threshold with a local notion of the motivic oscillation index.