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

2021/08/16 by Seung‐Jo Jung, In‐Kyun Kim, Jung, Seung-Jo +5
Mathematics · #Mathematical functions and polynomials #Meromorphic and Entire Functions #Advanced Differential Equations and Dynamical Systems

paper · pdf · doi:10.48550/arxiv.2108.07231

Abstract

For a holomorphic function f on a complex manifold X, the Briançon-Skoda exponent e^{\rm 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^{\rm BS}(f)≤ d_X (:=dim X) by Brian\c con-Skoda. We prove that e^{\rm BS}(f)≤[d_X-2\widetildeα_f]+1 with -\widetildeα_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^{\rm BS}(f)≤ d_X-2 in the case f^-1(0) has only rational singularities, that is, if \widetildeα_f>1.

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