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Briançon-Skoda exponents and the maximal root of reduced Bernstein-Sato polynomials

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Abstract

For a holomorphic function f on a complex manifold X, the Briançon-Skoda exponent \(e^{\mathrm{BS}}(f)\) is the smallest integer k with \(f^k\in (\partial f)\) (replacing X with a neighborhood of \(f^{-1}(0)\)), where \((\partial f)\) denotes the Jacobian ideal of f. It is shown that \(e^{\mathrm{BS}}(f)\le d_X\) \((:=\dim X)\) by Briançon-Skoda. We prove that \(e^\mathrm{BS}(f)\le [d_X-2{\widetilde{\alpha }}_f]+1\) with \(-{\widetilde{\alpha }}_f\) the maximal root of the reduced Bernstein-Sato polynomial \(b_f(s)/(s+1)\), assuming the latter exists (shrinking X if necessary). This implies for instance that \(e^{\mathrm{BS}}(f)\le d_X-2\) in the case \(f^{-1}(0)\) has only rational singularities, that is, if \({\widetilde{\alpha }}_f>1\).

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Authors and Affiliations

  1. Department of Mathematics Education, and Institute of Pure and Applied Mathematics, Jeonbuk National University, Jeonju, 54896, Korea

    Seung-Jo Jung

  2. Department of Mathematics, Yonsei University, Seoul, Korea

    In-Kyun Kim

  3. RIMS Kyoto University, Kyoto, 606-8502, Japan

    Morihiko Saito

  4. Department of Mathematics, Chungbuk National University, Cheongju-si, Chungcheongbuk-do, 28644, Korea

    Youngho Yoon

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  1. Seung-Jo Jung
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  2. In-Kyun Kim
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  4. Youngho Yoon
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Correspondence to In-Kyun Kim.

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This work was partially supported by NRF grant funded by the Korea government(MSIT) (the first author: NRF-2021R1C1C1004097, the second author: NRF-2020R1A2C4002510, and the fourth author: NRF-2020R1C1C1A01006782). The first author was partially supported by “Research Base Construction Fund Support Program” funded by Jeonbuk National University in 2020. The third author was partially supported by JSPS Kakenhi 15K04816.

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Jung, SJ., Kim, IK., Saito, M. et al. Briançon-Skoda exponents and the maximal root of reduced Bernstein-Sato polynomials. Sel. Math. New Ser. 28, 78 (2022). https://doi.org/10.1007/s00029-022-00791-1

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