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A remarkable subset of poles of the motivic zeta function

Nero Budur, Eduardo de Lorenzo Poza, Quan Shi, Huaiqing Zuo

TL;DR

The article identifies a combinatorially defined subset of poles of the motivic zeta function, termed remarkable numbers, determined by any log resolution through the vanishing orders and divisor intersections of the exceptional components. It introduces the intrinsic top-dimension zeta function $Z^{ ext{td}}_f(T)$ to study these poles via contact loci, proving that its poles are independent of the resolution and correspond to remarkable numbers; these poles then bound or control the poles of the classical motivic zeta function $Z^{ ext{mot}}_f(T)$. A key methodological advancement is the top-degree specialization in signed graded rings, enabling explicit, constructive pole calculations. The paper then provides a complete description for unibranch plane curves, showing that the remarkable numbers coincide with rupture-divisor data and tie directly to monodromy eigenvalues; broader classes are explored through a wealth of examples, including reality, quasi-ordinary, cone, and Newton non-degenerate cases. Overall, the work reframes aspects of the monodromy conjecture by isolating a robust, intrinsically defined set of poles whose study is accessible via classical singularity theory and related invariants.

Abstract

For any polynomial f with complex coefficients we find a remarkable subset of poles of the motivic zeta function. It is combinatorially determined by any log resolution and it admits an intrinsic interpretation in terms of contact loci of f. This uncovers a new, unexpected difficulty with proving the monodromy conjecture.

A remarkable subset of poles of the motivic zeta function

TL;DR

The article identifies a combinatorially defined subset of poles of the motivic zeta function, termed remarkable numbers, determined by any log resolution through the vanishing orders and divisor intersections of the exceptional components. It introduces the intrinsic top-dimension zeta function to study these poles via contact loci, proving that its poles are independent of the resolution and correspond to remarkable numbers; these poles then bound or control the poles of the classical motivic zeta function . A key methodological advancement is the top-degree specialization in signed graded rings, enabling explicit, constructive pole calculations. The paper then provides a complete description for unibranch plane curves, showing that the remarkable numbers coincide with rupture-divisor data and tie directly to monodromy eigenvalues; broader classes are explored through a wealth of examples, including reality, quasi-ordinary, cone, and Newton non-degenerate cases. Overall, the work reframes aspects of the monodromy conjecture by isolating a robust, intrinsically defined set of poles whose study is accessible via classical singularity theory and related invariants.

Abstract

For any polynomial f with complex coefficients we find a remarkable subset of poles of the motivic zeta function. It is combinatorially determined by any log resolution and it admits an intrinsic interpretation in terms of contact loci of f. This uncovers a new, unexpected difficulty with proving the monodromy conjecture.
Paper Structure (6 sections, 28 theorems, 61 equations, 2 figures)

This paper contains 6 sections, 28 theorems, 61 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminary material
  3. Top-degree specializations
  4. Top-dimension zeta functions
  5. Plane curves
  6. Examples

Key Result

Theorem 1.2

Let $X$ be an $n$-dimensional smooth complex irreducible variety and $f:X\to\mathbb A^1$ a regular function with non-empty zero locus. Then:

Figures (2)

  • Figure 1: Partition of the resolution graph of a unibranch plane curve into groups.
  • Figure 2: Two multibranch counterexamples to Theorem \ref{['thm:rmkble-unibranch']}.

Theorems & Definitions (67)

  • Definition 1.1
  • Theorem 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Definition 1.5
  • Lemma 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.11
  • ...and 57 more