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Conjectures of Igusa for  p ‐adic local zeta functions and of Denef and Loeser for topological local zeta functions assert that (the real part of) the poles of these local zeta functions are roots of the Bernstein‐Sato polynomials (that is, the  b ‐functions). We prove these conjectures for certain hyperplane arrangements, including the case of reduced hyperplane arrangements in three‐dimensional affine space.

On the local zeta functions and the b ‐functions of certain hyperplane arrangements