Around the motivic monodromy conjecture for non-degenerate hypersurfaces

Around the motivic monodromy conjecture for non-degenerate hypersurfaces

I will discuss my ongoing effort to comprehend, from a geometric viewpoint, the motivic monodromy conjecture for a "generic" complex multivariate polynomial $f$, namely any polynomial $f$ that is non-degenerate with respect to its Newton polyhedron. This conjecture, due to Igusa and Denef--Loeser, states that for every pole $s$ of the motivic zeta function associated to $f$, $\exp(2\pi is)$ is a "monodromy eigenvalue" associated to $f$. On the other hand, the non-degeneracy condition on $f$ ensures that the singularity theory of $f$ is governed, up to a certain extent, by faces of the Newton polyhedron of $f$. The extent to which the former is governed by the latter is one key aspect of the conjecture, and will be the main focus of my talk.