The Multivariable Strong Monodromy Conjecture for Plane Curves

Abstract

Let $F=(f_1,\dots ,f_r)$ be a tuple of holomorphic germs on a smooth complex germ, and let $B_{F,0}$ be its Bernstein--Sato ideal. We develop an iterated-residue obstruction showing that a nonzero coefficient-valued residue class on an SNC stratum forces the corresponding exact affine parameter to lie in $Z(B_{F,0})$. As applications, we prove that every maximal-order polar hyperplane of the local multivariable topological zeta function is contained in the Bernstein--Sato zero locus, and that the same holds for every actual polar hyperplane associated with a tuple of reduced plane curve germs. The latter proves the topological multivariable Strong Monodromy Conjecture for plane curves.