Upper Bounds for Roots of $b$-Functions, following Kashiwara and Lichtin

Abstract

By building on a method introduced by Kashiwara (Invent. Math. 38 (1976/77), 33–53) and refined by Lichtin (Ark. Mat. 27 (1989), 283–304), we give upper bounds for the roots of certain $b$-functions associated to a regular function $f$ in terms of a log resolution of singularities. As applications, we recover with more elementary methods a result of Budur and Saito (J. Algebraic Geom. 14 (2005), 269–282) describing the multiplier ideals of $f$ in terms of the $V$-filtration of $f$ and a result of the second-named author with Popa (Forum Math. Sigma 8 (2020), Paper No. e19, 41) giving a lower bound for the minimal exponent of $f$ in terms of a log resolution.