  We prove a descent theorem of nearby cycle formula for Newton non-degenerate
functions at the origin as well as its motivic version (without assuming the
convenience condition). This is used in some papers without any proof although
its proof is quite nontrivial because of the existence of coordinate
hyperplanes which is completely neglected in the literature about the descent
theorem. In the isolated singularity case, it implies some well-known formula
for the number of Jordan blocks of the Milnor monodromy with the theoretically
maximal size, using a standard estimate of weights. It also provides a proof of
a modified version of the Steenbrink conjecture on spectral pairs for
non-degenerate functions with simplicial Newton polytopes in the isolated
singularity case (which is false in the non-simplicial case).
