Subtlety of oscillation indices of oscillatory integrals of real analytic functions

TL;DR

This paper investigates the relationship between oscillation indices of oscillatory integrals and the real log canonical threshold for real analytic functions, highlighting cases of coincidence and exceptions, and questioning existing formulas.

Contribution

It clarifies the subtle relationship between oscillation indices and log canonical thresholds, identifying specific exceptional cases and potential inconsistencies in standard formulas.

Findings

Oscillation index is always negative and bounded below by the log canonical threshold.

In many cases, the oscillation index and the log canonical threshold coincide.

Exceptions occur in certain even-variable, lower-degree cases, challenging existing formulas.

Abstract

For a locally defined real analytic function, we study the relation between the oscillation index of oscillatory integrals and the real log canonical threshold. The former is always negative, and its absolute value is greater than or equal to the latter. They coincide very often, but there are certain exceptional cases even in the Newton nondegenerate convenient homogeneous case, for instance if the number of variables is even and smaller than the degree. This does not seem compatible with some standard formula in the literature, and there must be some error somewhere, although it does not seem easy to find it inside this paper.