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Resolution of singularities for $C^{\infty}$ functions and meromorphy of local zeta functions

Joe Kamimoto

Abstract

In this paper, we attempt to resolve the singularities of the zero variety of a $C^{\infty}$ function of two variables as much as possible by using ordinary blowings up. As a result, we formulate an algorithm to locally express the zero variety in the ``almost'' normal crossings form, which is close to the normal crossings form but may include flat functions. As an application, we investigate analytic continuation of local zeta functions associated with $C^{\infty}$ functions of two variables. As is well known, the desingularization theorem of Hironaka implies that the local zeta functions associated with real analytic functions admit the meromorphic continuation to the whole complex plane. On the other hand, it is recently observed that the local zeta function associated with a specific (non-real analytic) $C^{\infty}$ function has a singularity different from the pole. From this observation, the following questions are naturally raised in the $C^{\infty}$ case: how wide the meromorphically extendible region can be and what kinds of information essentially determine this region? This paper shows that this region can be described in terms of some kind of multiplicity of the zero variety of each $C^{\infty}$ function. By using our blowings up algorithm, it suffices to investigate local zeta functions in the almost normal crossings case. This case can be effectively analyzed by using real analysis methods; in particular, a van der Corput-type lemma plays a crucial role in the determination of the above region.

Resolution of singularities for $C^{\infty}$ functions and meromorphy of local zeta functions

Abstract

In this paper, we attempt to resolve the singularities of the zero variety of a function of two variables as much as possible by using ordinary blowings up. As a result, we formulate an algorithm to locally express the zero variety in the ``almost'' normal crossings form, which is close to the normal crossings form but may include flat functions. As an application, we investigate analytic continuation of local zeta functions associated with functions of two variables. As is well known, the desingularization theorem of Hironaka implies that the local zeta functions associated with real analytic functions admit the meromorphic continuation to the whole complex plane. On the other hand, it is recently observed that the local zeta function associated with a specific (non-real analytic) function has a singularity different from the pole. From this observation, the following questions are naturally raised in the case: how wide the meromorphically extendible region can be and what kinds of information essentially determine this region? This paper shows that this region can be described in terms of some kind of multiplicity of the zero variety of each function. By using our blowings up algorithm, it suffices to investigate local zeta functions in the almost normal crossings case. This case can be effectively analyzed by using real analysis methods; in particular, a van der Corput-type lemma plays a crucial role in the determination of the above region.
Paper Structure (46 sections, 36 theorems, 121 equations, 1 figure)

This paper contains 46 sections, 36 theorems, 121 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Description of the problems and the main result
  3. Newton data
  4. Holomorphic extension problem
  5. Meromorphic extension problem
  6. The quantity $\mu_0(f)$
  7. Main theorem
  8. Almost desingularization theorem for $C^{\infty}$ functions
  9. Almost resolution of singularities
  10. Construction of blowings up
  11. Blowing up of an open set in ${\mathbb C}^2$
  12. A series of blowings up
  13. A series of real blowings up
  14. Geometry of exceptional curves
  15. Rychkov's factorization formula and decisive curves
  16. ...and 31 more sections

Key Result

Theorem 2.2

${\mathfrak c}_0(f) (={\mathfrak h}_0(f))=1/\delta_0(f)$ holds for every non-flat $f\in C^{\infty}(U)$.

Figures (1)

  • Figure 1: In the case where $\mu_0(f)<\delta_0(f)$.

Theorems & Definitions (75)

  • Remark 2.1
  • Theorem 2.2: Gre06
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Definition 3.1
  • Theorem 3.2
  • Remark 3.3
  • Definition 3.4
  • ...and 65 more