Motivic-type Invariants of Blow-analytic Equivalence

To a given analytic function germ $f:({\mathbb{R}}^d,0)\to (\mathbb{R},0)$, we associate zeta functions $Z_{f,+}$, $Z_{f,-}\in \mathbb{Z}[[T]]$, 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.