Oscillation Index of Oscillatory Integrals

• Oscillation index is defined as the exponent that captures the decay rate of oscillatory integrals based on the local behavior of degenerate critical points.
• The Newton polyhedron and toric resolution techniques enable precise computation of the index for real-analytic, multidimensional phases.
• Exceptional phenomena, such as symmetry cancellation in even dimensions, require refined methods to achieve accurate asymptotic estimates.

An oscillatory integral is a complex-valued integral of the form

$$I(\lambda) = \int_{U} e^{i \lambda \varphi(x)} a(x) \, dx,$$

where $\lambda \to +\infty$ is a real parameter, $\varphi: U \to \mathbb{R}$ is the phase function, and $a$ is the amplitude. The oscillation index—which may also be called the leading oscillation exponent—quantifies the precise power-law decay rate of $I(\lambda)$ as $\lambda \to \infty$, and is fundamentally determined by analytic and geometric properties of the phase $\varphi$ near its degenerate critical points. The oscillation index governs estimates and full asymptotics in harmonic analysis, singularity theory, and related fields. Its computation and interpretation span one-dimensional model cases, multidimensional generalizations, and subtle connections with invariants such as the Newton polyhedron and the real log-canonical threshold.

1. Definition and Fundamental Properties

The oscillation index at a critical point is defined as the exponent $\alpha$ such that the integral $I(\lambda)$ admits, as $\lambda \to \infty$, an expansion of the form

$$I(\lambda) \sim C \lambda^{\alpha} (\log \lambda)^{m-1} + \text{lower order terms},$$

where $C$ is a nonzero constant for generic choice of amplitude. The value of $\alpha$ depends on the local behavior of the phase function at its critical points:

• In the one-variable case with an isolated degenerate critical point at $x=0$ and phase $\varphi(x)=x^p$, $p>1$, the oscillation index is $\mu=1/p$ so that $I(\lambda)\sim C\lambda^{-1/p}$ for suitable $a$ (Nagano et al., 2020).
• For real-analytic phases in several variables, the oscillation index is expressed in terms of the geometry of the Newton polyhedron $\Gamma_{+}(\varphi)$ of the phase: $\sigma(\varphi) = 1/d(\varphi)$ with $d(\varphi)$ the Newton distance, the minimal $t>0$ such that $(t,\ldots, t)\in \Gamma_{+}(\varphi)$ (Gilula, 2016).
• For general real-analytic $f$, the oscillation index $\alpha(f)$ is the maximal real exponent in the generalized asymptotic expansion of $I(\lambda)$, and always satisfies $\alpha(f)<0$ (Kim et al., 20 Nov 2025).

In the case of multiple critical points, the global oscillation index is the minimum over all local indices, corresponding to the slowest decaying contribution (Nagano et al., 2022).

2. One-Dimensional and Model Examples

For a one-dimensional phase $\varphi(x)=x^p$ with $p>1$, the asymptotic behavior is governed as follows (Nagano et al., 2020, Nagano et al., 2022):

• The critical point at $x=0$ is of "order $p$": $\varphi^{(j)}(0)=0$ for $1\le j < p$, $\varphi^{(p)}(0)\neq 0$.
• The decay rate is given by the change of variables $x=\lambda^{-1/p} y$, yielding $I(\lambda)\sim C\lambda^{-1/p}$.
• The full expansion is

$$I(\lambda) = \sum_{k=0}^{N-1} I_{p,k+1} \frac{a^{(k)}(0)}{k!} \lambda^{-(k+1)/p} + O(\lambda^{-(N+1)/p}),$$

where $$I_{p,k+1}=\frac{1}{p}e^{i\pi(k+1)/(2p)}\Gamma\left(\frac{k+1}{p}\right)$$ generalizes the Fresnel integral.

Thus, the oscillation index in this setting is precisely $\mu=1/p$, and the leading term is explicitly computable in terms of $p$, $\Gamma$-functions, and derivatives of the amplitude.

3. Higher-Dimensional Theory and Newton Polyhedron

In higher dimensions, the decay rate of oscillatory integrals with real-analytic phases satisfying Varchenko's nondegeneracy condition is governed by the Newton polyhedron (Gilula, 2016, Kim et al., 20 Nov 2025):

• The Newton polyhedron $\Gamma_+(\varphi)$ is the convex hull of $\{\alpha\in\mathbb{N}^n:c_\alpha\ne 0\}$.
• The Newton distance $d(\varphi)$ is the minimal $t>0$ with $(t,t,\ldots,t)\in\Gamma_+(\varphi)$.
• The oscillation index is $\sigma(\varphi)=1/d(\varphi)$, and the multiplicity $m$ is the codimension of the principal face $F_0$ which contains the diagonal point $(d(\varphi),\dots,d(\varphi))$.
• Under nondegeneracy, the sharp decay estimate is

$$|I(\lambda)|\le C\lambda^{-\sigma(\varphi)}(\log \lambda)^{m-1}.$$

• The entire sequence of exponents in the full expansion is determined combinatorially by the geometry of $\Gamma_+(\varphi)$, with logarithmic terms corresponding to faces of higher codimension.

The Newton polyhedron approach allows for extraction of the oscillation index and full asymptotics directly from the series expansion of the phase.

4. Comparisons with Real Log-Canonical Threshold

The oscillation index is closely related to the real log-canonical threshold (rlct), defined via the minimal exponent $lct_\mathbb{R}(f)>0$ for which $|f|^{-s}$ fails to be integrable near $0$ (Kim et al., 20 Nov 2025):

$$lct_\mathbb{R}(f) = \min_{j \in J} \frac{k_j+1}{m_j}$$

where $f$ is reduced (by resolution of singularities) to monomial form on local charts, and $k_j, m_j$ encode vanishing orders.

The general relationship is

$\alpha(f)\le -lct_\mathbb{R}(f).$

In the Newton-nondegenerate, convenient case (which includes generic homogeneous polynomials with nonvanishing principal part), and provided $lct_\mathbb{R}(f)<1$, equality holds:

$\alpha(f) = -lct_\mathbb{R}(f) = -\frac{1}{d(f)}.$

However, there are exceptional Newton-nondegenerate cases—when the number of variables $n$ is even and either $d>n$ or $d$ is even and $f^{-1}(0)=\{0\}$—where the candidate leading term vanishes, and strict inequality

$\alpha(f) < -lct_\mathbb{R}(f)$

holds (Kim et al., 20 Nov 2025).

5. Extensions: Product Phases and Singularities

For phases that are sums of monomials in each variable (separated, or product phases), the oscillation index is additive:

• For $\varphi(x)=\sum_{j=1}^n x_j^{m_j}$, the index is $\sigma=\sum_{j=1}^n \frac{1}{m_j}$, with explicit leading coefficient given as a product of generalized Fresnel integrals (Nagano et al., 2022).
• In the setting of "Arnold-type" singularities such as $A_k$, $E_6$, $E_8$,
  • $A_k$: $\varphi(x, y) = x^{k+1} + y^2$ yields index $1/(k+1) + 1/2$,
  • $E_6$: $\varphi(x, y) = x^3 + y^4$ yields index $1/3 + 1/4$,
  • $E_8$: $\varphi(x, y) = x^3 + y^5$ yields index $1/3 + 1/5$
  • with all coefficients and expansions computable in terms of one-dimensional contributions (Nagano et al., 2022).

6. Methodologies for Expansion and Computation

Two principal methodologies apply:

• Direct expansion and change of variables: For monomial-type phases, expansion of the amplitude near the critical point, followed by substitution $x=\lambda^{-1/p} y$, yields the asymptotic expansion, with each term corresponding to higher derivatives of the amplitude and explicit integral coefficients (Nagano et al., 2020).
• Newton polyhedron and toric modification: In multidimensional analytic settings, toric (or Hironaka) resolution replaces the phase by monomials (modulo units) on resolved charts, allowing the oscillation index and log-canonical threshold to be identified with minima over Newton data (Gilula, 2016, Kim et al., 20 Nov 2025).

In degenerate or analytic singularity settings, the full expansion incorporates logarithmic factors, combinatorially controlled by the principal face's codimension.

7. Exceptional Phenomena and Subtleties

While the oscillation index typically coincides (in absolute value) with the real log-canonical threshold in generic Newton-nondegenerate cases, recent work demonstrates subtle exceptions due to cancellation effects:

• When $n$ (number of variables) and $d$ (degree) are both even and $d>n$ or $f^{-1}(0)=\{0\}$, symmetry causes the candidate $\lambda^{-n/d}$ term to vanish; the decay rate is governed by the next-leading exponent (e.g., for $f(x, y)=x^4 + y^4$, $\alpha(f)=-1 <-1/2$) (Kim et al., 20 Nov 2025).

This highlights the limits of Newton nondegeneracy as a complete criterion and demonstrates the need to examine parity and multiplicity effects in leading-term computations.

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Key References:

• Generalized Fresnel integrals and one-variable expansions: (Nagano et al., 2020, Nagano et al., 2022).
• Multivariable and Newton polyhedron theory: (Gilula, 2016).
• Oscillation index versus real log-canonical threshold; exceptional phenomena: (Kim et al., 20 Nov 2025).