Analytic equivalence of normal crossing functions on a real analytic manifold (Q2894578)

The authors study the classification, up to real analytic equivalence, of real analytic functions having only normal crossing singularities (by the Hironaka desingularization theorem, any analytic function falls into this case after a finite sequence of blowing-ups with smooth centre). The first main result in the present paper states that for such functions defined on a real analytic manifold, \(C^\infty\) right equivalence implies analytic right equivalence. The proof consists in a careful use of Cartan's theorems A and B, and of Oka's theorem, in order to construct the analytic vector fields whose flows will yield analytic diffeomorphisms. The second main result establishes that in the case of compact analytic manifolds, the cardinality of the set of equivalence classes is zero or countable. Its proof uses a reduction to the Nash case, as well as a number of ingredients from the theory of Nash functions and manifolds.