# On Archimedean Zeta Functions and Newton Polyhedra

## TL;DR

This paper provides an explicit list of candidate poles for complex local zeta functions associated with non-degenerate polynomials, based on Newton polyhedra, refining previous results and extending them to the complex case.

## Contribution

It introduces a new explicit list of candidate poles for complex local zeta functions using Newton polyhedra, simplifying previous approaches and extending real case results to complex polynomials.

## Key findings

- List of candidate poles derived from Newton polyhedra
- Refinement of Varchenko's results for complex case
- Generalization of Denef and Sargos' results to complex setting

## Abstract

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.