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

Seung-Jo Jung, In-Kyun Kim, Morihiko Saito, Youngho Yoon

• For a holomorphic function f on a complex manifold X, the Briançon-Skoda exponent eBS( 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 eBS( f ) ≤ dX (:= dim X) by Briançon-Skoda. We prove that eBS( f ) ≤ [dX −2α 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 eBS( f ) ≤ dX −2 in the case f −1(0) has only rational singularities, that is, if α f > 1.