Poles of real motivic zeta functions for curves

TL;DR

This paper investigates the poles of real topological and motivic zeta functions associated with real polynomial functions in two variables, revealing their invariance under blow-Nash equivalence and extending prior complex case studies.

Contribution

It introduces the study of poles of real motivic and topological zeta functions for plane curves, connecting them to blow-Nash invariants and extending complex case results.

Findings

Identified the poles of real topological zeta functions for plane curves.

Established invariance of these poles under blow-Nash equivalence.

Extended the analysis of motivic zeta functions to the real setting.

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.