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

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