Motivic-type invariants of blow-analytic equivalence

Abstract

To a given analytic function germ f:(ℝ d ,0)→(ℝ,0), we associate zeta functions Z f,+ , Z f,- ∈ℤ[[T]], defined analogously to the motivic zeta functions of Denef and Loeser. We show that our zeta functions are rational and that they are invariants of the blow-analytic equivalence in the sense of Kuo. Then we use them together with the Fukui invariant to classify the blow-analytic equivalence classes of Brieskorn polynomials of two variables. Except special series of singularities our method classifies as well the blow-analytic equivalence classes of Brieskorn polynomials of three variables.