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Igusa's $p$-Adic Local Zeta Function and the Monodromy Conjecture for Non-Degenerate Surface Singularities (Memoirs of the American Mathematical Society)

Algebra

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About the Author Bart Bories and Willem Veys, Katholieke Universiteit Leuven, Belgium. Product Description 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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Format: Paperback
ASIN: 147041841X
Category: Books > Subjects > Science, Nature & Maths > Mathematics > Algebra
Domain: Amazon UK
Release Date: 30 June 2016
Listed Since: 06 May 2016

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Igusa's $p$-Adic Local Zeta Function and the Monodromy Conjecture for Non-Degenerate Surface Singularities (Memoirs of the American Mathematical Society)

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