On the monodromy conjecture for non-degenerate hypersurfaces
The authors verify the monodromy conjecture [\textit{J. Denef} and \textit{F. Loeser}, J. Am. Math. Soc. 5, No. 4, 705--720 (1992; Zbl 0777.32017)] about topological zeta functions for a large class of singularities that are non-degenerate with respect to their Newton polyhedra, including all such singularities of functions depending on four variables. They explain the novelty of their approach as follows. Thus, it is known that in the case of three variables all singularities close to a non-degenerate one are non-degenerate as well [\textit{A. Lemahieu} and \textit{L. Van Proeyen}, Trans. Am. Math. Soc. 363, No. 9, 4801--4829 (2011; Zbl 1248.14012)]. Next, in the setting of \textit{E. Artal Bartolo} et al. [Quasi-ordinary power series and their zeta functions. Providence, RI: American Mathematical Society (AMS) (2005; Zbl 1095.14005)], all singularities close to a quasi-ordinary one are also quasi-ordinary. However, in contrast to these papers, a new phenomenon arises in the four-dimensional case: there are degenerate singularities arbitrarily close to a nonisolated non-degenerate singularity; this is one of the main difficulties of the proof pointed out by the authors. The paper is divided into several parts. In the introduction, the authors give a detailed review of a number of previously obtained results and the corresponding extensive bibliography, explain the essence of their approach, and formulate the main statements. Then the monodromy conjecture for the topological zeta function and the main properties of Newton polyhedra are discussed, and configurations of faces of the Newton polyhedron that do not ensure the existence of the corresponding pole of the topological zeta function are studied. After that, the authors study face configurations, which, on the contrary, always nontrivially contribute to the multiplicity of the expected monodromy eigenvalue and prove the conjecture for a certain class of non-degenerate singularities in arbitrary dimension. The last section contains a proof of the monodromy conjecture for non-degenerate singularities of functions depending on four variables. In the appendix, it is briefly discussed some basic concepts related to the geometry of the lattice, and the necessary results used in the article.
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