The holomorphy conjecture for ideals in dimension two

Lemahieu, Ann; Van Proeyen, Lise

doi:10.1090/S0002-9939-2011-11186-9

The holomorphy conjecture for ideals in dimension two
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by Ann Lemahieu and Lise Van Proeyen

Proc. Amer. Math. Soc. 139 (2011), 3845-3852

DOI: https://doi.org/10.1090/S0002-9939-2011-11186-9

Published electronically: June 29, 2011

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Abstract:

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

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