Subtlety of oscillation indices of oscillatory integrals of real analytic functions

TL;DR

We investigate the relationship between the oscillation index $\beta(f)$ of real-analytic oscillatory integrals and the real log canonical threshold $rlct(f)$. Using resolution of singularities and Newton-polyhedral data, we relate $rlct$ to the invariant $\gamma(f)$ and derive sharp bounds on the oscillation index. Under the assumptions of Newton $\mathbb{R}$-nondegeneracy, convenience, and even $n$, the paper proves $\gamma(f)=n/d$ and $\beta(f)<-\gamma(f)$, implying $rlct(f)=n/d$. The work clarifies subtle distinctions between real and complex invariants and sharpens real-analytic oscillatory-integral asymptotics, with concrete criteria for when oscillation-index equalities hold or fail.

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 where the number of variables is even and smaller than the degree.

Theorems & Definitions (14)

• Theorem 1: rlct
• Theorem 2: V
• Theorem 3
• Remark 1.1
• Remark 1.2
• Remark 1.3
• Theorem 2.1: CKN
• Corollary 2.1
• proof
• Corollary 2.2