On the monodromy conjecture for non-degenerate hypersurfaces *

Recently the second author and Van Proeyen proved the monodromy conjecture on topological zeta functions for all non-degenerate surface singularities. In this paper, we obtain some higher-dimensional analogues of their results. First we study configurations of B 1-pyramid facets which produce fake poles. Secondly, we introduce fully supermodular functions which are useful to find eigenvalues of mon-odromies. Finally, we obtain a result which would be useful to treat the monodromy conjecture for non-degenerate hypersurfaces whose Newton polyhedron is not convenient. In particular, we prove the conjecture for non-degenerate hypersurface singularities in C 4 under some additional assumptions