Weighted homogeneous polynomials and blow-analytic equivalence

Abderrahmane, Ould M.

doi:10.2969/aspm/04310333

Cited by 3 publications

(4 citation statements)

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“…Remark 3•21. In [1], it is shown that the weights of a non-degenerate weighted homogeneous polynomial in two variable are invariants under blow-analytic equivalence, using the zeta function defined with the Euler characteristic of the homology of locally finite chains with closed supports [13]. Because of the good properties of the virtual Poincaré polynomial, we recover easily the analogous result, in the setting of blow-Nash equivalence.…”

Section: •3 Convenient Weighted Homogeneous Polynomialsmentioning

confidence: 57%

“…We express the measure of the arc spaces associated with a polynomial function f in terms of its Newton polyhedron. The set of integers k ∈ N * for which the arc space A k ( f ) is not empty has already been studied in the context of blow-analytic equivalence and is called the set of Fukui invariants [1,13]. It coincides with the set of exponents that appear in the zeta function of f with non-zero coefficients.…”

Section: •2 Motivic Invariant Of a Polynomial Functionmentioning

confidence: 99%

“…The question of the invariance of the weights for weighted homogeneous polynomial functions under blow-analytic equivalence already appeared as a conjecture in [9] and as a question in [13]. A positive answer has been given by O. M. Abderrahmane [1] in the two variable case, using two invariants of blow-analytic equivalence: Fukui invariants [9] and zeta functions [13] constructed by S. Koike and A. Parusiński using motivic integration [5] with the Euler characteristic of the homology of locally finite chains with closed supports as a measure.…”

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confidence: 99%

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Motivic invariants of real polynomial functions and their Newton polyhedrons

Fichou¹,

Fukui²

2015

Math. Proc. Camb. Phil. Soc.

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Section: •3 Convenient Weighted Homogeneous Polynomialsmentioning

confidence: 57%

Section: •2 Motivic Invariant Of a Polynomial Functionmentioning

confidence: 99%

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confidence: 99%

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Motivic invariants of real polynomial functions and their Newton polyhedrons

Fichou¹,

Fukui²

2015

Math. Proc. Camb. Phil. Soc.

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“…In the two variable case, the positive answer has been given by Abderrahmane for weighted homogeneous polynomials non‐degenerate with respect to their Newton polyhedron.…”

Section: Introductionmentioning

confidence: 99%

On the arc-analytic type of some weighted homogeneous polynomials

Campesato

2018

J. London Math. Soc.

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It is known that the weights of a complex weighted homogeneous polynomial f with isolated singularity are analytic invariants of (C d , f −1 (0)). When d = 2, 3 this result holds by assuming merely the topological type instead of the analytic one.Fichou and Fukui recently proved the following real counterpart: the blow-Nash type of a real singular non-degenerate convenient weighted homogeneous polynomial in three variables determines its weights.The aim of this paper is to generalize the above-cited result with no condition on the number of variables. We work with a characterization of the blow-Nash equivalence called the arc-analytic equivalence. It is an equivalence relation on Nash function germs with no continuous moduli which may be seen as a semialgebraic version of the blow-analytic equivalence of Kuo. Contents

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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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Weighted homogeneous polynomials and blow-analytic equivalence

Cited by 3 publications

References 3 publications

Motivic invariants of real polynomial functions and their Newton polyhedrons

Motivic invariants of real polynomial functions and their Newton polyhedrons

On the arc-analytic type of some weighted homogeneous polynomials

Blow-analytic equivalence of two variable real analytic function germs

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