Blow-analytic Equivalence of Two Variable Real Analytic Function Germs

Blow-analytic equivalence is a notion for real analytic function germs, introduced by Tzee-Char Kuo in order to develop the real analytic equisingularity theory. In this paper we give several complete characterisations of blow-analytic equivalence in the two dimensional case in terms of the minimal resolutions, the real tree model for the arrangement of Newton-Puiseux roots, and the cascade blow-analytic equivalence. These characterisations show that in the two dimensional case the blow-analytic equivalence is the right real analogue of the topological equivalence of complex analytic function germs. In addition, in the general n-dimensional case, we show that a real modification in the sense of Kuo satisfies the arc-lifting property. In a search for a ”right” equivalence relation of real analytic function germs, that could play a similar role to the topological equivalence in the complex analytic set-up, at the end of 1970 Tzee-Char Kuo proposed the notion of blow-analytic equivalence [19, 20, 21, 23, 24, 25, 26, 27]). In [27], Kuo proved that blow-analytic equivalence is an equivalence relation and established a local finiteness of blow-analytic types for analytic families of real analytic function-germs with isolated singularities. Apart from this result, many blowanalytic triviality theorems are shown and several blow-analytic invariants are introduced, as we shall mention below. In this paper we give a complete characterisation of blow-analytic equivalence classes of two variable real analytic function germs. Theorem 0.1. Let f : (R, 0) → (R, 0) and g : (R, 0) → (R, 0) be real analytic function germs. Then the following conditions are equivalent: (1) f and g are blow-analytically equivalent. (2) f and g have weakly isomorphic minimal resolution spaces. (3) The real tree models of f and g are isomorphic. Theorem 0.1 can be stated in both the oriented and non-oriented cases, see section 8 below. By a weak isomorphism of resolution spaces we mean a homeomorphism that preseves the basic numerical data of resolutions, see subsection 1.3 below. Remark 0.2. A classical result of Zariski [31] shows that the topological type of a complex analytic function germ (C, 0) → (C, 0) is completely characterised by the Puiseux pairs of the irreducible components of the zero set, their multiplicities, and the intersection 1991 Mathematics Subject Classification. Primary: 32S15. Secondary: 14B05.