the residue eigenvalues of these Gauß-Manin systems then need to be symmetric around zero. This was stated as a conjecture in loc.cit., and it follows from the relation between these eigenvalues and the roots of the Bernstein polynomial that we prove here.

Finally, the family of Brieskorn lattices associated to $f_{|D_t}$ also has logarithmic extension over the divisor $D$, constructed using logarithmic differential forms. We define the fibre over $t=0$ to be the *logarithmic Brieskorn lattice* of $D$ (in fact, it does not depend on the choice of the linear form). In contrast to the fibres at $t\neq 0$, this Brieskorn lattice is regular singular at the origin, reflecting the local situation of the pair $(f,h)$ at the origin in ${{\mathds C}}^n$. It turns out that then our result can be rephrased to give the analogue of Malgrange’s classical result for isolated singularities: The roots of the Bernstein polynomial are (up to a rescaling) the residue eigenvalues on the *saturation* of this logarithmic Brieskorn lattice.

**Acknowledgements:** I thank Claude Sabbah and Michel Granger for their help during the preparation of this article and Mathias Schulze for comments on a first version. I am particularly grateful to Ignacio de Gregorio for 