On the monodromy conjecture for non-degenerate hypersurfaces

TL;DR

This paper proves the Denef–Loeser conjecture for a broad class of non-degenerate hypersurface singularities in multiple variables, including four-variable cases, by developing new analytical tools for complex multidimensional phenomena.

Contribution

It establishes the conjecture for non-degenerate hypersurfaces in multiple variables and introduces new methods to handle degenerate singularities close to non-degenerate ones.

Findings

Proved the conjecture for non-degenerate hypersurfaces of four variables.

Developed new tools for analyzing multidimensional singularities.

Identified the challenges posed by degenerate singularities near non-degenerate ones.

Abstract

The monodromy conjecture is an umbrella term for several conjectured relationships between poles of zeta functions, monodromy eigenvalues and roots of Bernstein-Sato polynomials in arithmetic geometry and singularity theory. Even the weakest of these relations -- the Denef--Loeser conjecture on topological zeta functions -- is open for surface singularities. We prove it for a wide class of multidimensional singularities that are non-degenerate with respect to their Newton polyhedra, including all such singularities of functions of four variables. A crucial difference from the known case of three variables is the existence of degenerate singularities arbitrarily close to a non-degenerate one. Thus, even aiming at the study of non-degenerate singularities, we have to go beyond this setting. We develop new tools to deal with such multidimensional phenomena, and conjecture how the proof for non-degenerate singularities of arbitrarily many variables might look like.