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On a suspension formula for Denef-Loeser zeta functions

E. Artal Bartolo, P. D. González Pérez, M. González Villa, E. León-Cardenal

Abstract

Formulas for the topological zeta functions of suspensions by 2 points are due to Artal et al. We generalize these formulas to the motivic level and for arbitrary suspensions, by using a stratification principle and classical techniques of generating functions in toric geometry. The same strategy is used to obtain formulas for the motivic zeta functions of some families of non-isolated singularities related to superisolated, Lê-Yomdin, and weighted Lê-Yomdin singularities.

On a suspension formula for Denef-Loeser zeta functions

Abstract

Formulas for the topological zeta functions of suspensions by 2 points are due to Artal et al. We generalize these formulas to the motivic level and for arbitrary suspensions, by using a stratification principle and classical techniques of generating functions in toric geometry. The same strategy is used to obtain formulas for the motivic zeta functions of some families of non-isolated singularities related to superisolated, Lê-Yomdin, and weighted Lê-Yomdin singularities.

Paper Structure

This paper contains 12 sections, 13 theorems, 96 equations.

Table of Contents

  1. General settings and notations
  2. Arithmetic functions
  3. Arc spaces
  4. Grothendieck rings and their specializations
  5. Denef-Loeser zeta functions
  6. General setting
  7. Notations
  8. Naive and motivic Zeta functions for certain binomials
  9. Topological Zeta functions for certain binomials
  10. Computation of the local motivic zeta functions of G
  11. A general formula for the topological zeta functions of G and F
  12. Comparison with previous work

Key Result

Lemma 1.2

The Jordan's totient functions $J_k$ and the functions $\sigma_k$ are related under Dirichlet convolution by the following expressions or, equivalently, by the expressions

Theorems & Definitions (30)

  • Remark 1.1
  • Lemma 1.2
  • Proposition 1.3: ACNLM-ASENS
  • Lemma 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Lemma 3.1
  • ...and 20 more