Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Poles of real motivic zeta functions for curves

Théo Jaudon

Abstract

To a given real polynomial function f $\in$ R[x1, . . . , x d ], we associate real topological zeta functions Ztop,0(f\,; s) and Z $\pm$ top,0 (f\,; s) $\in$ Q(s), analogous to the topological zeta function of Denef and Loeser in the complex case. These functions are specializations of the real motivic zeta functions studied in [Fic05a] and [Cam17]. Therefore, these functions and their sets of poles are invariants of the blow-Nash equivalence. Using the approach of [Vey95], we study the poles of these real topological zeta functions, as well as real motivic zeta functions, when f is a real polynomial in two variables.

Poles of real motivic zeta functions for curves

Abstract

To a given real polynomial function f R[x1, . . . , x d ], we associate real topological zeta functions Ztop,0(f\,; s) and Z top,0 (f\,; s) Q(s), analogous to the topological zeta function of Denef and Loeser in the complex case. These functions are specializations of the real motivic zeta functions studied in [Fic05a] and [Cam17]. Therefore, these functions and their sets of poles are invariants of the blow-Nash equivalence. Using the approach of [Vey95], we study the poles of these real topological zeta functions, as well as real motivic zeta functions, when f is a real polynomial in two variables.
Paper Structure (7 sections, 33 theorems, 169 equations, 13 figures)

This paper contains 7 sections, 33 theorems, 169 equations, 13 figures.

Table of Contents

  1. Real motivic zeta functions
  2. Poles of the real naive zeta function for curves
  3. Study of a contribution
  4. The real dual graph of a resolution
  5. Poles of zeta functions with signs for curves
  6. Study of a contribution : the virtual Poincaré polynomial of a real curve of hyperelliptic type
  7. Poles and monodromy eigenvalues

Key Result

Theorem 1

For $a \in \{f=0\}$, with $0 < \delta \ll \varepsilon \ll 1$, let $D_{\delta}^* \subset \mathbb{C}$ denote the punctured open disc of radius $\delta$ and let $B_{a,\varepsilon} \subset \mathbb{C}^d$ be the closed ball centered at $a$ with radius $\varepsilon$. The restriction is a locally trivial smooth fibration. Its fiber is denoted by $\mathcal{F}_{f,a}$ and is called the Milnor fiber of $f$ a

Figures (13)

  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • Figure :
  • ...and 8 more figures

Theorems & Definitions (87)

  • Theorem 1: milnor1
  • Conjecture 2: Monodromy conjecture, weak version
  • Theorem 3: veys1 Theorem 4.3
  • Theorem 4: = Theorem \ref{['thm 4']}
  • Theorem 5: = Corollary \ref{['cor 6']}
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Definition 1.5
  • ...and 77 more