, Volume 26, Issue 2, pp 753-772
Date: 22 May 2012

Zeta functions and Bernstein–Sato polynomials for ideals in dimension two

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Abstract

For a nonzero ideal \(\mathcal {I}\lhd \mathbf {C}[x_{1},\ldots,x_{n}]\) , with \(0\in \operatorname {supp}\mathcal {I}\) , a generalization of a conjecture of Igusa–Denef–Loeser predicts that every pole of its topological zeta function is a root of its Bernstein–Sato polynomial. However, typically only a few roots are obtained this way. Following ideas of Veys (Adv. Math. 213(1):341–357, 2007), we study the following question. Is it possible to find a collection \(\mathcal{G}\) of polynomials gC[x 1,…,x n ], such that, for all \(g\in\mathcal{G}\) , every pole of the topological zeta function associated to \(\mathcal {I}\) and the volume form gdx 1∧⋯∧dx n on the affine n-space, is a root of the Bernstein–Sato polynomial of \(\mathcal {I}\) , and such that all roots are realized in this way. We obtain a negative answer to this question, providing counterexamples for monomial and principal ideals in dimension two, and give a partial positive result as well.