roots in concrete examples. This is true even for linear free divisors, though the differential operator occurring in Bernstein’s functional equation is more explicitly known than in the general case. On the other hand, the calculation of the residue eigenvalues alluded to above is, although not trivial, easier to carry out. We apply our result to obtain, using the calculations from [@dGMS chapter 6], Bernstein polynomials for discriminants in representation spaces of the Dynkin quivers $A_n$, $D_n$ and $E_6$ as well as the so-called star quiver $\star_n$, also considered in loc.cit. (which is not a Dynkin quiver for $n>3$). We also calculate the Bernstein polynomials for two irreducible linear free divisors which are discriminants of irreducible pre-homogenous vector spaces described in [@sk].

Another motivation for this work comes from the fact that the residue eigenvalues of the Gauß-Manin systems give information on their limit behavior (resp., of the corresponding family of Brieskorn lattices), when approaching the zero fibre of $h$, i.e., the divisor $D$. In particular, in [@dGMS], questions about the degeneration of *Frobenius manifolds*, associated to the tame functions $f_{|D_t}$, where $D_t:=h^{-1}(t)$, were related to the asymptotic behavior of a natural pairing defined on the Gauß-Manin system. In particular, 