The holomorphy conjecture for ideals in dimension two

Abstract

The holomorphy conjecture predicts that the topological zeta function associated to a polynomial f ∈ C [ x 1 , … , x n ] f \in \mathbb {C}[x_1,\ldots ,x_n] and an integer d > 0 d > 0 is holomorphic unless d d divides the order of an eigenvalue of local monodromy of f f . In this paper, we generalise the holomorphy conjecture to the setting of arbitrary ideals in C [ x 1 , … , x n ] \mathbb {C}[x_1,\ldots ,x_n] , and we prove it when n = 2 n=2 .