On a motivic invariant of the arc-analytic equivalence

TL;DR

This paper introduces a new motivic zeta function for Nash function germs that is invariant under arc-analytic equivalence, enabling classification of certain polynomials through new convolution and Thom-Sebastiani formulas.

Contribution

It defines a novel motivic zeta function with richer structure, providing new formulas and invariants for arc-analytic equivalence classification.

Findings

The zeta function generalizes previous constructions with additional structure.

It satisfies a convolution formula and a Thom-Sebastiani type formula.

Exponents of Brieskorn polynomials are shown to be arc-analytic invariants.

Abstract

To a Nash function germ, we associate a zeta function similar to the one introduced by J. Denef and F. Loeser. Our zeta function is a formal power series with coefficients in the Grothendieck ring $\mathcal{M}$ of $\mathcal{AS}$-sets up to $\mathbb{R}^*$-equivariant $\mathcal{AS}$-bijections over $\mathbb{R}^*$, an analog of the Grothendieck ring constructed by G. Guibert, F. Loeser and M. Merle. This zeta function generalizes the previous construction of G. Fichou but thanks to its richer structure it allows us to get a convolution formula and a Thom-Sebastiani type formula. We show that our zeta function is an invariant of the arc-analytic equivalence, a version of the blow-Nash equivalence of G. Fichou. The convolution formula allows us to obtain a partial classification of Brieskorn polynomials up to the arc-analytic equivalence by showing that the exponents are arc-analytic invariants.