[Paper Review] Coefficients of the poles of local zeta functions and their applications to oscillating integrals

Toshihisa Okada, Kiyoshi Takeuchi|ArXiv.org|Mar 25, 2009

Analytic Number Theory Research14 references3 citations

TL;DR

This paper introduces a novel method to precisely compute the coefficients of poles in local zeta functions associated with real analytic functions with isolated singularities, using meromorphic continuation of distributions. It derives explicit formulas for these coefficients and applies them to oscillating integrals, yielding a closed-form expression for the leading asymptotic coefficient involving gamma functions and complex phases.

ABSTRACT

We introduce a new method which enables us to calculate the coefficients of the poles of local zeta functions very precisely and prove some explicit formulas. Some vanishing theorems for the candidate poles of local zeta functions will be also given. Moreover we apply our method to oscillating integrals and obtain an explicit formula for the coefficients of their asymptotic expansions.

Motivation & Objective

• To develop a precise method for calculating coefficients of poles in local zeta functions, which had been largely unaddressed in general settings.
• To provide explicit formulas for the coefficients of the Laurent expansion of local zeta functions at their poles.
• To apply these results to oscillating integrals and derive an explicit formula for the leading coefficient in their asymptotic expansion.
• To establish vanishing theorems for coefficients in the asymptotic expansion based on the jet structure of the test function.
• To connect the coefficients of the zeta function poles with those of oscillating integrals via complex analysis and resolution of singularities.

Proposed method

• Utilizes meromorphic continuation of the distribution $ |x_{1+}^{l_1 au + m_1} imes ext{...} imes x_{n+}^{l_n au + m_n}| $ with respect to complex parameter $ au $, enabling precise pole coefficient analysis.
• Employs a resolution of singularities via a smooth subdivision $ au $ of the dual fan $ au_0 $ of the Newton polygon $ \Gamma_+(f) $, focusing on cones $ \sigma \in \Sigma $.
• Defines key invariants: $ l(a^i(\sigma)) = \min_{\alpha \in \Gamma_+(f)} \langle a^i(\sigma), \alpha \rangle \in \mathbb{Z}_+ $ and $ |a^i(\sigma)| = \sum_j a^i(\sigma)_j $, used to characterize candidate poles.
• Introduces sets $ \Sigma_j^{(k)} $ of cones where a candidate pole $ -\lambda_j $ lies in the set $ K^i(\sigma) $, enabling classification of pole orders.
• Defines $ \nu(\sigma)_i \in \mathbb{Z}_+ $ such that $ \lambda_j = \frac{|a^i(\sigma)| + \nu(\sigma)_i}{l(a^i(\sigma))} $, crucial for computing Laurent coefficients.
• Uses the distributional derivative and Taylor expansion of $ \varphi $ at 0, combined with $ \partial^{\mu(\sigma,\alpha)} |f_\sigma|^{-\lambda_j}(0) $, to express coefficients explicitly.

Experimental results

Research questions

• RQ1 How can the coefficients of the poles of local zeta functions be computed with high precision in a general setting?
• RQ2 What conditions ensure the vanishing of coefficients in the asymptotic expansion of oscillating integrals?
• RQ3 What is the explicit formula for the leading coefficient $ c_{j,n}(\varphi) $ in the asymptotic expansion of $ I_f(\varphi)(t) $?
• RQ4 How do the coefficients of the zeta function poles relate to those of oscillating integrals?
• RQ5 Under what conditions does the deepest pole $ \lambda = -\lambda_j $ contribute to the asymptotic expansion with logarithmic terms?

Key findings

• The coefficient $ a_{j,n}^{\pm}(\varphi) $ of the deepest pole $ \lambda = -\lambda_j $ in $ Z_f^{\pm}(\varphi) $ is given by a sum over $ \sigma \in \Sigma_j^{(n)} $, involving $ c_{\pm}(\sigma) $, factorials of multi-indices $ \mu(\sigma,\alpha) $, and derivatives of $ |f_\sigma|^{-\lambda_j} $ at 0.
• When $ \lambda_j $ is not an integer, $ a_{j,n}^{\pm}(\varphi) = b_{j,n}^{\pm}(\varphi) $, establishing a direct link between zeta and oscillating integral coefficients.
• The leading coefficient $ c_{j,n}(\varphi) $ in the asymptotic expansion of $ I_f(\varphi)(t) $ is $ \frac{\Gamma(\lambda_j)}{(n-1)!} \left( e^{\frac{\pi i}{2}\lambda_j} b^+_{j,n}(\varphi) + e^{-\frac{\pi i}{2}\lambda_j} b^-_{j,n}(\varphi) \right) $, explicitly relating oscillating integrals to zeta function coefficients.
• Vanishing theorems are proven: if the jet of $ \varphi $ at 0 avoids certain multi-indices $ \Delta_{j,k} $, then $ c_{j,k}(\varphi) = \cdots = c_{j,k_j}(\varphi) = 0 $, providing a criterion for coefficient vanishing.
• The formula for $ b_{j,n}^{\pm}(\varphi) $ involves a sum over $ \sigma \in \Sigma_j^{(n)} $, with terms weighted by $ \prod_{i=1}^n \frac{1}{l(a^i(\sigma)) \mu(\sigma,\alpha)_i!} $, and includes the $ \mu(\sigma,\alpha) $-th derivative of $ |f_\sigma|^{-\lambda_j} $ at 0.
• The identity $ c_+(\sigma) + c_-(\sigma) = \prod_{i=1}^n \{1 + (-1)^{\nu(\sigma)_i}\} $ holds for each $ \sigma \in \Sigma^{(n)}_j $, linking the sign-dependent coefficients to the pole structure.