Around the motivic monodromy conjecture for non-degenerate hypersurfaces

Abstract

We provide a new, geometric proof of the motivic monodromy conjecture for non-degenerate hypersurfaces in dimension $3$, which has been proven previously by the work of Lemahieu--Van Proeyen and Bories--Veys. More generally, given a non-degenerate complex polynomial $f$ in any number of variables and a set $\mathbf{B}$ of $B_1$-facets of the Newton polyhedron of $f$ with consistent base directions, we construct a stack-theoretic embedded desingularization of $f^{-1}(0)$ above the origin, whose set of numerical data excludes any known candidate pole of the motivic zeta function of $f$ at the origin that arises solely from facets in $\mathbf{B}$. We anticipate that the constructions herein might inspire new insights as well as new possibilities towards a solution of the conjecture.