Motivic Milnor Fibres in Families of Real Singularities

Abstract

In this article we prove two results concerning the motivic Milnor fibres $S^{\epsilon }(f)$ associated to a map germ $f:({\mathbb{R}}^n,0)\to (\mathbb{R},0)$, defined by G. Comte and G. Fichou. Firstly, we prove that if $f,g:({\mathbb{R}}^n,0)\to (\mathbb{R},0)$ are arc-analytically equivalent germs of Nash functions then the virtual Poincaré polynomial of the corresponding motivic Milnor fibres are equal. This extends (and provides a new proof) of a result of G. Fichou. Secondly, let $T\subset {\mathbb{R}}^m$ be a real algebraic set and $f:T\times ({\mathbb{R}}^n,0)\to \mathbb{R}$ a polynomial function of polynomial map germs such that $f(t,0)=0$ for any $t\in T$. Then we prove that there exists a locally finite real analytic stratification $\mathcal{S}$ of $T$ such that if $S\in \mathcal{S}$ is a stratum then $f_t{\sim }_a{f}_{t^{\prime }}$ are arc-analytically equivalent, for any $t,t^{\prime }\in S$. Furthermore, if $T$ is compact then the stratification $\mathcal{S}$ can be taken to be finite. These two results imply in particular that the virtual Poincaré polynomials of the respective zeta functions are equal i.e $\beta (S^{\epsilon }(f_t))=\beta (S^{\epsilon }(f_{t^{\prime }}))$.

This is part of my thesis completed in 2020 at LAMA