In 2011 Lemahieu and Van Proeyen proved the Monodromy Conjecture for the local topological zeta function of a non-degenerated surface singularity. We start from their work and obtain the same result for Igusa's p-adic and the motivic zeta function. In the p-adic case, this is, for a polynomial f ā Z[x, y, z] satisfying f (0, 0, 0) = 0 and non-degenerated with respect to its Newton polyhedron, we show that every pole of the local p-adic zeta function of f induces an eigenvalue of the local monodromy of f at some point of f ā1 (0) ā C 3 close to the origin.Essentially the entire paper is dedicated to proving that, for f as above, certain candidate poles of Igusa's p-adic zeta function of f , arising from socalled B 1 -facets of the Newton polyhedron of f , are actually not poles. This turns out to be much harder than in the topological setting. The combinatorial proof is preceded by a study of the integral points in three-dimensional fundamental parallelepipeds. Together with the work of Lemahieu and Van Proeyen, this main result leads to the Monodromy Conjecture for the p-adic and motivic zeta function of a non-degenerated surface singularity. p 97 10. The main theorem in the motivic setting 101 10.1. The local motivic zeta function and the motivic Monodromy Conjecture 101 10.2. A formula for the local motivic zeta function of a non-degenerated polynomial 104 10.3. A proof of the main theorem in the motivic setting 113 References 118(3)Suppose that the cone Ī“ i is strictly positively spanned by the linearly independent primitive vectors v j , j ā J i , in Z n 0 \ {0}. Then we have S(Ī“ i )(s) = Ī£(Ī“ i )(s) jāJi (p Ļ(vj )+m(vj )s ā 1), 13 The theorem was announced in [12] and a proof is written down in [13].
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