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OverviewIn 2011 Lemahieu and Van Proeyen proved the Monodromy Conjecture for the local topological zeta function of a non-degenerate surface singularity. The authors 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 $f\in\mathbf{Z}[x,y,z]$ satisfying $f(0,0,0)=0$ and non-degenerate 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 $f^{-1}(0)\subset\mathbf{C}^3$ 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 so-called $B_1$-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-degenerate surface singularity. Full Product DetailsAuthor: Bart Bories , Willem VeysPublisher: American Mathematical Society Imprint: American Mathematical Society Weight: 0.218kg ISBN: 9781470418410ISBN 10: 147041841 Pages: 131 Publication Date: 30 June 2016 Audience: Professional and scholarly , Professional & Vocational Format: Paperback Publisher's Status: Active Availability: Out of stock  The supplier is temporarily out of stock of this item. It will be ordered for you on backorder and shipped when it becomes available. Table of ContentsChapter 1. Introduction Chapter 2. On the Integral Points in a Three-Dimensional Fundamental Parallelepiped Spanned by Primitive Vectors Chapter 3. Case I: Exactly One Facet Contributes to s0s0 and this Facet Is a B1B1-Simplex Chapter 4. Case II: Exactly One Facet Contributes to s0s0 and this Facet Is a Non-Compact B1B1-Facet Chapter 5. Case III: Exactly Two Facets of ?f?f Contribute to s0s0, and These Two Facets Are Both B1B1-Simplices with Respect to a Same Variable and Have an Edge in Common Chapter 6. Case IV: Exactly Two Facets of ?f?f Contribute to s0s0, and These Two Facets Are Both Non-Compact B1B1-Facets with Respect to a Same Variable and Have an Edge in Common Chapter 7. Case V: Exactly Two Facets of ?f?f Contribute to s0s0; One of Them Is a Non-Compact B1B1-Facet, the Other One a B1B1-Simplex; These Facets Are B1B1 with Respect to a Same Variable and Have an Edge in Common Chapter 8. Case VI: At Least Three Facets of ?f?f Contribute to s0s0; All of Them Are B1B1-Facets (Compact or Not) with Respect to a Same Variable and They Are ’Connected to Each Other by Edges’ Chapter 9. General Case: Several Groups of B1B1-Facets Contribute to s0s0; Every Group Is Separately Covered By One of the Previous Cases, and the Groups Have Pairwise at Most One Point in Common Chapter 10. The Main Theorem for a Non-Trivial c Character of Z×pZp× Chapter 11. The Main Theorem in the Motivic SettingReviewsAuthor InformationBart Bories and Willem Veys, Katholieke Universiteit Leuven, Belgium. Tab Content 6Author Website:Countries AvailableAll regions |
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