  Let $f$ be a polynomial function over the complex numbers and let $\phi$ be a
smooth function over $\mathbb{C}$ 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
$\phi$. 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.
