The monodromy conjecture for hyperplane arrangements
Let \(f\in\mathbb{C}[x_1,\dots,x_n]\) be a non-constant polynomial with \(f(0)=0\). The topological monodromy conjecture stated in [\textit{J. Denef} and \textit{F. Loeser}, Astérisque No. 201--203, Exp. No. 741, 359--386 (1991; Zbl 0777.32017)] relates the poles of the local topological zeta-function of \(f\) to eigenvalues of the monodromy on the cohomology of the Milnor fibre and to roots of its Bernstein-Sato polynomial. If \(f\) defines a hyperplane arrangement in \(\mathbb{C}^n\), the authors prove the first relation and reduce the proof of the second one to a weaker conjecture. They also prove the weaker conjecture for some special cases of hyperplane arrangements, in particular for those which are of moderate type. Furthermore, they recall the \(p\)-adic version of the monodromy conjecture given in [\textit{J.-I. Igusa}, An introduction to the theory of local zeta functions. AMS/IP Studies in Advanced Mathematics. 14. Providence, RI: American Mathematical Society (AMS) (2000; Zbl 0959.11047)] and prove a similar result in the case of \(\mathbb{Q}\)-hyperplane arrangements.
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