Dedicated to professor Hironaka in his 80th birthday.

Let f be a polynomial function over the complex numbers and let φ be a smooth function over C with compact support. When f is non-degenerate with respect to its Newton polyhedron, we give an explicit list of candidate poles for the complex local zeta function attached to f and φ. The provided list is given just in terms of the normal vectors to the supporting hyperplanes of the Newton polyhedron attached to f . More precisely, our list does not contain the candidate poles coming from the additional vectors required in the regular conical subdivision of the first orthant, and necessary in the study of local zeta functions through resolution of singularities.Our results refine the corresponding results of Varchenko and generalize the results of Denef and Sargos in the real case, to the complex setting.

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In this article we focus on the study of special parabolic points in surfaces arising as graphs of polynomials, we give a theorem of Viro's patchworking type to build families of real polynomials in two variables with a prescribed number of special parabolic points in their graphs. We use this result to build a family of degree d real polynomials in two variables with (d−4)(2d−9) special parabolic points in its graph. This brings the number of special parabolic points closer to the upper bound of (d − 2)(5d − 12) when d ≥ 13, which is the best known up until now.