Introduction to the monodromy conjecture

2024/03/05 by Willem Veys, Veys, Willem · 1 citation
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2403.03343

openalex publication_date 2024/03/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The monodromy conjecture is a mysterious open problem in singularity theory. Its original version relates arithmetic and topological/geometric properties of a multivariate polynomial f over the integers, more precisely, poles of the p-adic Igusa zeta function of f should induce monodromy eigenvalues of f. The case of interest is when the zero set of f has singular points. We first present some history and motivation. Then we expose a proof in the case of two variables, and partial results in higher dimension, together with geometric theorems of independent interest inspired by the conjecture. We conclude with several possible generalizations.

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• On the monodromy conjecture for determinantal varieties

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