$D$-Modules Generated by Rational Powers of Holomorphic Functions

We prove some suffcient conditions in order that a root of the Bernstein–Sato polynomial contributes to difference between certain $D$-modules generated by rational powers holomorphic function; for instance, this holds case isolated singularities with semisimple Milnor monodromies. then construct an example where does not contribute difference. This also solves old open problem about relation monodromy and exponential residue Gauss–Manin connection on saturation Brieskorn lattice. shows structure lattices can be more complicated than one might imagine.