Local Zeta Function for Curves, Non Degeneracy Conditions and Newton Polygons

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dc.creatorSaia, M. J.
dc.creatorZuniga-Galindo, W. A.
dc.date2001-07-26
dc.date2003-07-03
dc.date.accessioned2026-07-07T04:42:44Z
dc.date.available2026-07-07T04:42:44Z
dc.descriptionThis paper is dedicated to the description of the poles of the Igusa local zeta functions $Z(s,f,v)$ when $f(x,y)$ satisfies a new non degeneracy condition, that we have called arithmetic non degeneracy. More precisely, we attach to each polynomial $f(x,y)$, a collection of convex sets $Γ^{A}$ called the arithmetic Newton polygon of $f(x,y)$, and introduce the notion of arithmetic non degeneracy with respect to $Γ^{A}(f)$. The set of degenerate polynomials, with respect to a fixed geometric Newton polygon, contains an open subset, for the Zariski topology, formed by non degenerate polynomials with respect to some arithmetic Newton polygon.If $L$ is a number field, our main result asserts that for almost all non-archimedean valuations $v$ of $L$, the poles of $Z(s,f,v),$ with $f(x,y)\in L[x,y]$, can be described explicitly in terms of the equations of the straight segments that conform the boundaries of the convex sets that belong to $Γ^{A}(f)$. Moreover, our proof gives an effective procedure to compute $Z(s,f,v)$.
dc.description32 pages, 4 figures
dc.identifierhttps://arxiv.org/abs/math/0107189
dc.identifierhttp://arxiv.org/abs/math/0107189
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/61909
dc.subjectAlgebraic Geometry
dc.subjectNumber Theory
dc.subject11D79, 14G20,14M25
dc.titleLocal Zeta Function for Curves, Non Degeneracy Conditions and Newton Polygons
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