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Introduction to the Monodromy Conjecture

In: Handbook of Geometry and Topology of Singularities VII

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  • Willem Veys

    (KU Leuven, Departement Wiskunde)

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.

Suggested Citation

  • Willem Veys, 2025. "Introduction to the Monodromy Conjecture," Springer Books, in: José Luis Cisneros-Molina & Lê Dũng Tráng & José Seade (ed.), Handbook of Geometry and Topology of Singularities VII, chapter 0, pages 721-765, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-68711-2_13
    DOI: 10.1007/978-3-031-68711-2_13
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