  Given a family $X$ of complex varieties degenerating over a punctured disc,
one is interested in computing related invariants called the motivic nearby
fiber and the refined limit mixed Hodge numbers, both of which contain
information about the induced action of monodromy on the cohomology of a fiber
of $X$. Our first main result is that the motivic nearby fiber of $X$ can be
computed by first stratifying $X$ into locally closed subvarieties that are
non-degenerate in the sense of Tevelev, and then applying an explicit formula
on each piece of the stratification that involves tropical geometry. Our second
main result is an explicit combinatorial formula for the refined limit mixed
Hodge numbers in the case when $X$ is a family of non-degenerate hypersurfaces.
As an application, given a complex polynomial, then, under appropriate
conditions, we give a combinatorial formula for the Jordan block structure of
the action of monodromy on the cohomology of the Milnor fiber, generalizing a
famous formula of Varchenko for the associated eigenvalues. In addition, we
give a formula for the Jordan block structure of the action of monodromy at
infinity.
