Around the motivic monodromy conjecture for non-degenerate hypersurfaces

TL;DR

This paper offers a new geometric proof of the motivic monodromy conjecture for non-degenerate hypersurfaces in three dimensions, extending previous results and providing a novel desingularization approach that excludes certain candidate poles.

Contribution

It introduces a stack-theoretic embedded desingularization method for non-degenerate hypersurfaces, advancing the understanding of the motivic monodromy conjecture in higher dimensions.

Findings

Proves the motivic monodromy conjecture for dimension 3 hypersurfaces.

Constructs a desingularization that excludes specific candidate poles.

Provides a framework potentially applicable to broader cases of the conjecture.

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.