Motivic-type invariants of blow-analytic equivalence. (Q1878467)

In this difficult and interesting work the authors develop techniques that allow to study and distinguish different blow-analytic classes of analytic function germs \(f:(\mathbb R^n,0)\to (\mathbb R,0)\). They adapt and apply to the real analytic set-up the ideas coming from motivic integrations, in particular the concept of motivic zeta functions due to \textit{J. Denef} and \textit{F. Loeser} [J. Algebr. Geom. 7, No.3, 505--537 (1998; Zbl 0943.14010); Invent. Math. 135, 201--232 (1999; Zbl 0928.14004); Duke Math. J. 99, No. 2, 285--309 (1999; Zbl 0966.14015)]. The blow-analytic equivalence is interesting because it does not allow continuous moduli for families of isolated singularities and it preserves a deep information on the algebraic structure of the singularity. On the other hand for real singularities, unlike for the complex ones, the topological classification is very weak. The blow-analytic equivalence was proposed by \textit{T. C. Kuo} [Invent. Math. 82, 257--262 (1985; Zbl 0587.32018)] in order to overcome this problem. Moreover the authors suggest by some examples that the blow-up analytic equivalence of real analytic function germs behaves in a similar way to the topological equivalence in the complex case. Until now there were very few results allowing to distinguish different blow-analytic types and hence to attempt a classification even for simplest analytic singularities. The main result of the paper is to introduce new invariants and to start such a classification.