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

Seung Jo Jung; In Kyun Kim; Morihiko Saito; Youngho Yoon

doi:10.1007/s00029-022-00791-1

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

Seung Jo Jung
, In Kyun Kim^*
, Morihiko Saito
, Youngho Yoon

^*Corresponding author for this work

Department of Mathematics Education

Yonsei University
Kyoto University
Chungbuk National University

Research output: Contribution to journal › Journal article › peer-review

3 Scopus citations

Abstract

For a holomorphic function f on a complex manifold X, the Briançon-Skoda exponent e^BS(f) is the smallest integer k with f^k∈ (∂f) (replacing X with a neighborhood of f^{- 1}(0)), where (∂f) denotes the Jacobian ideal of f. It is shown that e^BS(f) ≤ d_X(: = dim X) by Briançon-Skoda. We prove that e^BS(f) ≤ [d_X- 2 α~ _f] + 1 with - α~ _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^BS(f) ≤ d_X- 2 in the case f^{- 1}(0) has only rational singularities, that is, if α~ _f> 1.

Original language	English
Article number	78
Journal	Selecta Mathematica, New Series
Volume	28
Issue number	4
DOIs	https://doi.org/10.1007/s00029-022-00791-1
State	Published - 2022.09

Quacquarelli Symonds(QS) Subject Topics

Mathematics
Physics & Astronomy

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@article{a1078b9175164f848e7fd20d6cbc75f2,

title = "Brian{\c c}on-Skoda exponents and the maximal root of reduced Bernstein-Sato polynomials",

abstract = "For a holomorphic function f on a complex manifold X, the Brian{\c c}on-Skoda exponent eBS(f) is the smallest integer k with fk∈ (∂f) (replacing X with a neighborhood of f- 1(0)), where (∂f) denotes the Jacobian ideal of f. It is shown that eBS(f) ≤ dX(: = dim X) by Brian{\c c}on-Skoda. We prove that eBS(f) ≤ [dX- 2 α\textasciitilde{} f] + 1 with - α\textasciitilde{} f the maximal root of the reduced Bernstein-Sato polynomial bf(s) / (s+ 1) , assuming the latter exists (shrinking X if necessary). This implies for instance that eBS(f) ≤ dX- 2 in the case f- 1(0) has only rational singularities, that is, if α\textasciitilde{} f> 1.",

author = "Jung, \{Seung Jo\} and Kim, \{In Kyun\} and Morihiko Saito and Youngho Yoon",

note = "Publisher Copyright: {\textcopyright} 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.",

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doi = "10.1007/s00029-022-00791-1",

language = "English",

volume = "28",

journal = "Selecta Mathematica, New Series",

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TY - JOUR

T1 - Briançon-Skoda exponents and the maximal root of reduced Bernstein-Sato polynomials

AU - Jung, Seung Jo

AU - Kim, In Kyun

AU - Saito, Morihiko

AU - Yoon, Youngho

PY - 2022/9

Y1 - 2022/9

N2 - For a holomorphic function f on a complex manifold X, the Briançon-Skoda exponent eBS(f) is the smallest integer k with fk∈ (∂f) (replacing X with a neighborhood of f- 1(0)), where (∂f) denotes the Jacobian ideal of f. It is shown that eBS(f) ≤ dX(: = dim X) by Briançon-Skoda. We prove that eBS(f) ≤ [dX- 2 α~ f] + 1 with - α~ f the maximal root of the reduced Bernstein-Sato polynomial bf(s) / (s+ 1) , assuming the latter exists (shrinking X if necessary). This implies for instance that eBS(f) ≤ dX- 2 in the case f- 1(0) has only rational singularities, that is, if α~ f> 1.

AB - For a holomorphic function f on a complex manifold X, the Briançon-Skoda exponent eBS(f) is the smallest integer k with fk∈ (∂f) (replacing X with a neighborhood of f- 1(0)), where (∂f) denotes the Jacobian ideal of f. It is shown that eBS(f) ≤ dX(: = dim X) by Briançon-Skoda. We prove that eBS(f) ≤ [dX- 2 α~ f] + 1 with - α~ f the maximal root of the reduced Bernstein-Sato polynomial bf(s) / (s+ 1) , assuming the latter exists (shrinking X if necessary). This implies for instance that eBS(f) ≤ dX- 2 in the case f- 1(0) has only rational singularities, that is, if α~ f> 1.

UR - https://www.scopus.com/pages/publications/85137591631

U2 - 10.1007/s00029-022-00791-1

DO - 10.1007/s00029-022-00791-1

M3 - Journal article

AN - SCOPUS:85137591631

SN - 1022-1824

VL - 28

JO - Selecta Mathematica, New Series

JF - Selecta Mathematica, New Series

IS - 4

M1 - 78

ER -

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

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