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

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This 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 Γᵃ called the arithmetic Newton polygon of f(x,y), and introduce the notion of arithmetic non degeneracy with respect to Γᵃ(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)∈ 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 Γᵃ(f). Moreover, our proof gives an effective procedure to compute Z(s,f,v).