Motivic-type invariants of blow-analytic equivalence

Koike, Satoshi; Parusiński, Adam

doi:10.5802/aif.2001

Cited by 26 publications

(63 citation statements)

References 31 publications

(71 reference statements)

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“…In the latter case, additionally, f (x, y) = x p − y q and g(x, y) = x p + y q are blow analytically equivalent, cf. [18], but they are not analytically equivalent. Moreover, f and g are not bi-lipschitz equivalent.…”

Section: Cascade Blow-analytic Homeomorphisms and Their Geometric Promentioning

confidence: 99%

“…We complete the general case in Corollary 2.5 below. Apart from the Fukui invariants, motivic type invariants are introduced in [18] and [5].…”

Section: Invariants Of Blow-analytic Equivalencementioning

confidence: 99%

“…The blow-analytic (non-oriented) classification of Brieskorn two variable singularities ±x p ± y q was obtained in [18] using the Fukui invariants and the zeta functions. This classification coincides with (x, y) → (±x, ±y) classification except for the case p odd and q = pm with m even.…”

Section: Cascade Blow-analytic Homeomorphisms and Their Geometric Promentioning

confidence: 99%

“…Invariants allowing to distinguish different blow-analytic types were constructed, using the geometry of arc-spaces and motivic integration, by Fukui [7], Fichou [5], and in [18]. These constructions are based on the observation that blow-analytic homeomorphisms send real analytic arcs to real analytic arcs.…”

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confidence: 99%

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Blow-analytic equivalence of two variable real analytic function germs

Koike¹,

Parusiński²

2009

J. Algebraic Geom.

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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.

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Section: Cascade Blow-analytic Homeomorphisms and Their Geometric Promentioning

confidence: 99%

“…We complete the general case in Corollary 2.5 below. Apart from the Fukui invariants, motivic type invariants are introduced in [18] and [5].…”

Section: Invariants Of Blow-analytic Equivalencementioning

confidence: 99%

Section: Cascade Blow-analytic Homeomorphisms and Their Geometric Promentioning

confidence: 99%

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confidence: 99%

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Blow-analytic equivalence of two variable real analytic function germs

Koike¹,

Parusiński²

2009

J. Algebraic Geom.

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“…The latter has already been studied with slightly different definitions (see in particular S. Koike & A. Parusiński [11] and T. Fukui & L. Paunescu [7]). Roughly speaking, two given real analytic function germs are blow-analytic equivalent if they are topologicaly equivalent and moreover, after suitable modifications, they become analytically equivalent.…”

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confidence: 99%

Zeta functions and blow-Nash equivalence

Fichou

2005

Ann. Polon. Math.

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Abstract.We propose a refinement of the notion of blow-Nash equivalence between Nash function germs, which has been introduced in [2] as an analog in the Nash setting of the blow-analytic equivalence defined by T.-C. Kuo [13]. The new definition is more natural and geometric. Moreover, this equivalence relation still does not admit moduli for a Nash family of isolated singularities. But though the zeta functions constructed in [2] are no longer invariants for this new relation, thanks to a Denef & Loeser formula coming from motivic integration in a Nash setting, we manage to derive new invariants for this equivalence relation.

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On Grothendieck rings and algebraically constructible functions

Fichou

2017

Math. Ann.

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Motivic-type invariants of blow-analytic equivalence

Cited by 26 publications

References 31 publications

Blow-analytic equivalence of two variable real analytic function germs

Blow-analytic equivalence of two variable real analytic function germs

Zeta functions and blow-Nash equivalence

On Grothendieck rings and algebraically constructible functions

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