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Project

Igusa type zeta functions and the monodromy conjecture

For a polynomial f over the integers, there are important results and open questions relating its arithmetic and geometric properties. The famous monodromy conjecture is such an open problem. Here the arithmetic part consists of numbers of solutions of congruences, that is, solutions of the equation f=0 over the integers, modulo some given number. Technically, one gathers this information in a generating series, called the Igusa zeta function. The geometric part consists of the monodromy eigenvalues of f, which are invariants of the solutions of f=0 over the complex numbers, more precisely related to the non-smooth or singular points of this solution set. The conjecture predicts a precise link between these two a priori unrelated packages of information. Our main project consists in studying this conjecture, especially for polynomials in three variables. As a side project, we apply techniques from the study of Igusa zeta functions to questions in group theory.

Date:1 Jan 2018 →  31 Dec 2021
Keywords:Zeta functions, Monodromy conjecture, Igusa type
Disciplines:Analysis, Applied mathematics in specific fields, General mathematics, History and foundations, Other mathematical sciences and statistics