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Complete classification of Brieskorn polynomials up to the arc-analytic equivalence

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Abstract

It has been recently proved that the arc-analytic type of a singular Brieskorn polynomial determines its exponents. This last result may be seen as a real analogue of a theorem by Yoshinaga and Suzuki concerning the topological type of complex Brieskorn polynomials. In the real setting it is natural to investigate further by asking how the signs of the coefficients of a Brieskorn polynomial change its arc-analytic type. The aim of the present paper is to answer this question by giving a complete classification of Brieskorn polynomials up to the arc-analytic equivalence. The proof relies on an invariant of this relation whose construction is similar to the one of Denef–Loeser motivic zeta functions. The classification obtained generalizes the one of Koike–Parusiński in the two variable case up to the blow-analytic equivalence and the one of Fichou in the three variable case up to the blow-Nash equivalence.

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Notes

  1. i.e. Polynomials of the form \(\pm x^p\pm y^q\).

  2. i.e. Polynomials of the form \(\pm x^p\pm y^q\pm z^r\).

  3. A Nash function is a smooth function with semialgebraic graph. Such a function is necessarily real analytic.

  4. This is a notion due to Kurdyka [17].

  5. Indeed, Kurdyka proved that a semialgebraic arc-analytic map is real analytic outside a set of codimension at least 2 [17, Théorème 5.2].

  6. A subset S of a real analytic manifold M is arc-symmetric if given a real analytic arc on M, either this arc is entirely included in S or it meets S at isolated points only. This is a notion due to Kurdyka [17].

  7. When \(k_i\) is odd, we could also have fixed that \(\varepsilon _i=1\).

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Acknowledgements

I am sincerely grateful to Adam Parusiński for our fruitful discussions during the preparation of this article. I express my gratitude and thanks to Toshizumi Fukui who warmly welcomed me in Saitama University where this work has been carried out.

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  1. Department of Mathematics, Faculty of Science, Saitama University, 255 Shimo-Okubo, Sakura-ku, Saitama, 338-8570, Japan

    Jean-Baptiste Campesato

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Correspondence to Jean-Baptiste Campesato.

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Research supported by a Japan Society for the Promotion of Science (JSPS) Postdoctoral Fellowship (Short-term) for North American and European Researchers.

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Campesato, JB. Complete classification of Brieskorn polynomials up to the arc-analytic equivalence. Math. Z. 290, 1145–1163 (2018). https://doi.org/10.1007/s00209-018-2056-7

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