Blow-analytic equivalence of two variable real analytic function germs

Abstract

Abstract. Blow-analytic equivalence is a notion for real analytic function germs, introduced by Tzee-Char Kuo in order to develop real analytic equisingularity theory. In this paper we give complete characterisations of blow-analytic equivalence in the two dimensional case: in terms of the real tree model for the arrangement of real parts of Newton-Puiseux roots and their Puiseux pairs, and in terms of minimal resolutions. These characterisations show that in the two dimensional case the blow-analytic equivalence is a natural analogue of topological equivalence of complex analytic function germs. Moreover, we show that in the two-dimensional case the blow-analytic equivalence can be made cascade, and hence satisfies several geometric properties. It preserves, for instance, the contact orders of real analytic arcs.In the general n-dimensional case, we show that a singular real modification satisfies the arc-lifting property.A classical result of Burau [4] and Zariski [34] shows the embedded topological type of a plane curve singularity (X, 0) ⊂ (C 2 , 0) is determined by the Puiseux pairs of each irreducible component and the intersection numbers of any pairs of distinct components. It can be shown, cf. [30], that the topological type of function germs f : (C 2 , 0) → (C, 0) is completely characterised, also in the non-reduced case f = f In this paper we give a real analytic counterpart of these results and show that the two variable version of blow-analytic equivalence of Kuo is classified by invariants similar to Puiseux pairs, multiplicities of irreducible components, and intersection numbers. Moreover we show several natural geometric properties of this equivalence, answering previously posed questions. In the main result of this paper we a give complete characterisation of blow-analytic equivalence classes of two variable real analytic function germs.Theorem 0.1. Let f : (R 2 , 0) → (R, 0) and g : (R 2 , 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.Moreover if f and g are blow-analytically equivalent then they are equivalent by a cascadeblow-analytic homeomorphism.1991 Mathematics Subject Classification. Primary: 32S15. Secondary: 14B05.

Equivalence relations for two variable real analytic function germs

“…By [13] f and g are not blow-analytically equivalent by an orientation preserving blow-analytic homeomorphism. We construct below an orientation preserving bi-Lipschitz homeomorphism σ :…”

“…In this section we present several examples of bi-Lipschitz equivalent real analytic function germs that are not blow-analytically equivalent. In order to distinguish different blow-analytic types we use either the real tree model of [13] or the Fukui invariants. Recall the definition of Fukui invariants of blow-analytic equivalence, c.f.…”

“…f (x, y) = x(x 3 − y 5 )(x 3 + y 5 ), g(x, y) = x(x 3 − ay 5 )(x 3 − by 5 ), where 0 < a < b are constants. The real trees of f and g are not equivalent, see below, hence by [13], f and g are not blow-analytically equivalent. The polynomial P has two non-degenerate critical points −1 < z 1 < 0, z 2 = −z 1 and P (z 1 ) > 0, P (z 2 ) = −P (z 1 ) < 0.…”

“…Thus, there is no direct relation between these two notions. Nevertheless, as shown in [13], a blow-analytic homeomorphism that gives blow-analytic equivalence between two 2-variable real analytic function germs, preserves the order of contact between non-parameterised real analytic arcs. Note that by the curve selection lemma, a subanalytic homeomorphism is bi-Lipschitz if and only if it preserves the order of contact between parameterised real analytic arcs.…”

“…We say that a homeomorphism germ h : (R n , 0) → (R n , 0) is a blow-analytic homeomorphism if there exist real modifications µ : (M, µ −1 (0)) → (R n , 0), μ : ( M , μ−1 (0)) → (R n , 0) and an analytic isomorphism Φ : (M, µ −1 (0)) → ( M , μ−1 (0)) so that σ•µ = μ•Φ. The formal definition of real modification is somewhat technical and, since in this paper we consider only the two variable case, we shall use the following criterion of [13]: in two variable case µ is a real modification if and only if it is a finite composition of point blowings-up. Finally, we say that two real analytic function germs f : (R n , 0) → (R, 0) and g : (R n , 0) → (R, 0) are blow-analytically equivalent if there exists a blow-analytic homeomorphism σ : (R n , 0) → (R n , 0) such that f = g • σ.…”