| eBook ISBN: | 978-1-4704-2944-7 |
| Product Code: | MEMO/242/1145.E |
| List Price: | $84.00 |
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| AMS Member Price: | $50.40 |
| eBook ISBN: | 978-1-4704-2944-7 |
| Product Code: | MEMO/242/1145.E |
| List Price: | $84.00 |
| MAA Member Price: | $75.60 |
| AMS Member Price: | $50.40 |
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 242; 2016; 131 ppMSC: Primary 14; 11; Secondary 52; 32; 58
In 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.
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Table of Contents
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Chapters
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1. Introduction
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2. On the Integral Points in a Three-Dimensional Fundamental Parallelepiped Spanned by Primitive Vectors
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3. Case I: Exactly One Facet Contributes to $s_0$ and this Facet Is a $B_1$-Simplex
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4. Case II: Exactly One Facet Contributes to $s_0$ and this Facet Is a Non-Compact $B_1$-Facet
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5. Case III: Exactly Two Facets of $\Gamma _f$ Contribute to $s_0$, and These Two Facets Are Both $B_1$-Simplices with Respect to a Same Variable and Have an Edge in Common
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6. Case IV: Exactly Two Facets of $\Gamma _f$ Contribute to $s_0$, and These Two Facets Are Both Non-Compact $B_1$-Facets with Respect to a Same Variable and Have an Edge in Common
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7. Case V: Exactly Two Facets of $\Gamma _f$ Contribute to $s_0$; One of Them Is a Non-Compact $B_1$-Facet, the Other One a $B_1$-Simplex; These Facets Are $B_1$ with Respect to a Same Variable and Have an Edge in Common
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8. Case VI: At Least Three Facets of $\Gamma _f$ Contribute to $s_0$; All of Them Are $B_1$-Facets (Compact or Not) with Respect to a Same Variable and They Are ’Connected to Each Other by Edges’
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9. General Case: Several Groups of $B_1$-Facets Contribute to $s_0$; Every Group Is Separately Covered By One of the Previous Cases, and the Groups Have Pairwise at Most One Point in Common
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10. The Main Theorem for a Non-Trivial Character of $\mathbf {Z}_p^{\times }$
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11. The Main Theorem in the Motivic Setting
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Additional Material
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In 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.
-
Chapters
-
1. Introduction
-
2. On the Integral Points in a Three-Dimensional Fundamental Parallelepiped Spanned by Primitive Vectors
-
3. Case I: Exactly One Facet Contributes to $s_0$ and this Facet Is a $B_1$-Simplex
-
4. Case II: Exactly One Facet Contributes to $s_0$ and this Facet Is a Non-Compact $B_1$-Facet
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5. Case III: Exactly Two Facets of $\Gamma _f$ Contribute to $s_0$, and These Two Facets Are Both $B_1$-Simplices with Respect to a Same Variable and Have an Edge in Common
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6. Case IV: Exactly Two Facets of $\Gamma _f$ Contribute to $s_0$, and These Two Facets Are Both Non-Compact $B_1$-Facets with Respect to a Same Variable and Have an Edge in Common
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7. Case V: Exactly Two Facets of $\Gamma _f$ Contribute to $s_0$; One of Them Is a Non-Compact $B_1$-Facet, the Other One a $B_1$-Simplex; These Facets Are $B_1$ with Respect to a Same Variable and Have an Edge in Common
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8. Case VI: At Least Three Facets of $\Gamma _f$ Contribute to $s_0$; All of Them Are $B_1$-Facets (Compact or Not) with Respect to a Same Variable and They Are ’Connected to Each Other by Edges’
-
9. General Case: Several Groups of $B_1$-Facets Contribute to $s_0$; Every Group Is Separately Covered By One of the Previous Cases, and the Groups Have Pairwise at Most One Point in Common
-
10. The Main Theorem for a Non-Trivial Character of $\mathbf {Z}_p^{\times }$
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11. The Main Theorem in the Motivic Setting
