Monodromy Jordan blocks, \(b\)-functions and poles of zeta functions for germs of plane curves (Q615828)

The paper is concerned with three local zeta functions: the Igusa zeta function and the topological and motivic zeta functions associated to an analytic function germ \(f\) in two variables. It was known that each of these three zeta functions have at most a double pole, given by the log canonical threshold of the corresponding plane curve singularity. When the germ \(f\) is reduced, Loeser showed that such a double pole always creates a Jordan block of size 2 in the monodromy of \(f\). The authors discuss this phenomenon in the case of a non-reduced germ \(f\) and obtain new information on the Bernstein-Sato polynomial in this situation, confirming a conjecture of Igusa, Denef and Loeser.