On the non-analyticity locus of an arc-analytic function

Kurdyka, Krzysztof; Parusiński, Adam

doi:10.1090/s1056-3911-2011-00553-5

Cited by 6 publications

(9 citation statements)

References 16 publications

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“…1.1], there exists a finite composition of blowings-up σ :Ř → R (with smooth Nash centres) which converts the arc-analytic semialgebraic function f • π| E into a Nash function f • π • σ|Ě, where the Nash manifoldĚ is the strict transform of E by σ. Moreover, by [15,Thm. 1.3], the centres of the blowings-up in σ can be chosen so that σ is an isomorphism outside the preimage of S(f • π).…”

Section: Proof Of Theoremmentioning

confidence: 85%

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Extensions of arc-analytic functions

Adamus

Seyedinejad

2018

Math. Ann.

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Section: Proof Of Theoremmentioning

confidence: 85%

“…Given f ∈ A a (X), let S(f ) denote the locus of points x ∈ Reg k (X) such that f is not analytic at x. Then, S(f ) is semialgebraic and dim S(f ) ≤ k − 2 (see [15], and cf. [13,Thm.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Extensions of arc-analytic functions

Adamus

Seyedinejad

2018

Math. Ann.

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“…In order to define “equivariant” generalizations of the zeta functions for Nash germs, we use an additive invariant defined on all - sets, that is Boolean combinations of arc-symmetric sets (see [16] and [17]) equipped with an algebraic action of : the equivariant virtual Poincaré series. It is defined in [9] using the equivariant virtual Betti numbers, which are the unique additive invariant on - sets coinciding with the dimensions of their equivariant homology.…”

Section: Equivariant Zeta Functionsmentioning

confidence: 99%

“…He showed, in particular, that blow-Nash equivalence is an equivalence relation and that it has no moduli for Nash families with isolated singularities. Using as a motivic measure the virtual Poincaré polynomial of McCrory and Parusiński in [20], extended to the wider category of sets [16] and [17] by Fichou in [7], one can generalize the zeta functions of Koike and Parusiński in [13]. In [8], Fichou showed that these latter zeta functions are invariants for blow-Nash equivalence via blow-Nash isomorphisms.…”

Section: Introductionmentioning

confidence: 99%

Equivariant Zeta Functions for Invariant Nash Germs

Priziac¹

2016

Nagoya Math. J.

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To any Nash germ invariant under right composition with a linear action of a finite group, we associate its equivariant zeta functions, inspired from motivic zeta functions, using the equivariant virtual Poincaré series as a motivic measure. We show Denef-Loeser formulae for the equivariant zeta functions and prove that they are invariants for equivariant blow-Nash equivalence via equivariant blow-Nash isomorphisms. Equivariant blow-Nash equivalence between invariant Nash germs is defined as a generalization involving equivariant data of the blow-Nash equivalence. IntroductionA crucial issue in the study of real analytic germs is the choice of a good equivalence relation by which we can distinguish them. One may think about C r -equivalence, r = 0, 1, . . . , ∞, ω. However, the topological equivalence seems, unlike the complex case, not fine enough : for example, all the germs of the form x 2m +y 2n are topologically equivalent. On the other hand, the C 1 -equivalence has already moduli : consider the Whitney family f t (x, y) = xy(y − x)(y − tx), t > 1, then f t and f t ′ are C 1 -equivalent if and only if t = t ′ . In [15], T.-C. Kuo proposed an equivalence relation for real analytic germs named the blow-analytic equivalence for which, in particular, analytically parametrized family of isolated singularities have a locally finite classification. Roughly speaking, two real analytic germs are said blow-analytically equivalent if they become analytically equivalent after composition with real modifications (e.g., finite successions of blowings-up along smooth centers). With respect to this equivalence relation, Whitney family has only one equivalence class. Slightly stronger versions of blow-analytic equivalence have been proposed so far, by S. Koike and A. Parusiński in [13] and T. Fukui and L. Paunescu in [8] for example. An important feature of blow-analytic equivalence is also that we have invariants for this equivalence relation, like the Fukui invariants ([7]) and the zeta functions ([13]) inspired by the motivic zeta functions of J. Denef and F. Loeser ([5]) using the Euler characteristic with compact supports as a motivic measure.The present paper is interested in the study of Nash germs, that is real analytic germs with semialgebraic graph. In [10], G. Fichou defined an analog adapted to Nash germs of the

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“…For example, an arc-analytic function on R n need not be continuous [5] or subanalytic [17], and its nonanalyticity locus can be nondiscrete even if n = 2 [18]. However, arc-analytic functions that are also semialgebraic proved to be very useful in real algebraic and analytic geometry (see [1,2,4,8,12,13,15,19,20,21] and the references therein). By [16,Proposition 5.1], every semialgebraic arc-analytic function is continuous.…”

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confidence: 99%

Arc-meromorphous functions

Kucharz

Kurdyka

2021

Ann. Polon. Math.

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We introduce arc-meromorphous functions, which are continuous functions representable as quotients of semialgebraic arc-analytic functions, and develop the theory of arc-meromorphous sheaves on Nash manifolds. Our main results are Cartan's theorems A and B for quasi-coherent arc-meromorphous sheaves. 0. Introduction. In this note, building on the theory of arc-analytic functions initiated by the second named author [16], we introduce arcmeromorphous functions and arc-meromorphous sheaves on Nash manifolds. Arc-meromorphous functions are analogs for regulous and Nash regulous functions studied in [8] and [13], respectively. The term "regulous" is derived from "regular" and "continuous", whereas "meromorphous" comes from "meromorphic" and "continuous". Our theory of arc-meromorphous sheaves is developed in parallel to the theories of regulous sheaves [8] (see also the recent survey [14]) and Nash regulous sheaves [13]. It is established in [8] and [13] that Cartan's theorems A and B hold for quasi-coherent regulous sheaves and quasi-coherent Nash regulous sheaves. Our main results are Theorem 2.4 (Cartan's theorem A) and Theorem 2.5 (Cartan's theorem B) for quasi-coherent arc-meromorphous sheaves. Recall that Cartan's theorems A and B fail for coherent real algebraic sheaves [6, Example 12.1.5], [7, Theorem 1] and coherent Nash sheaves [11]. We refer to [6] for the general theory of semialgebraic sets, semialgebraic functions, and related concepts. Recall that a Nash manifold is an analytic submanifold X ⊂ R n , for some n, which is also a semialgebraic set. A realvalued function on X is called a Nash function if it is both analytic and semialgebraic. By [22, Theorem VI.2.1, Remark VI.2.11], each Nash manifold is Nash isomorphic to a nonsingular algebraic set in R m , for some m.

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On the non-analyticity locus of an arc-analytic function

Cited by 6 publications

References 16 publications

Extensions of arc-analytic functions

Extensions of arc-analytic functions

Equivariant Zeta Functions for Invariant Nash Germs

Arc-meromorphous functions

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