Pole structure of Shintani zeta functions and Newton polytopes

TL;DR

This paper investigates the pole structure of Shintani zeta functions, revealing their relation to Newton polytopes and providing an algorithmic approach to understanding their hyperplanes.

Contribution

It refines the understanding of pole locations of Shintani zeta functions by linking them to Newton polytopes and introduces an algorithm for weight distribution on graphs.

Findings

Poles lie on hyperplanes parallel to facets of Newton polytopes.

Coefficients of hyperplane equations are zero or one.

An algorithm for weight distribution satisfying vertex bounds is proposed.

Abstract

It is known that Shintani zeta functions, which generalise multiple zeta functions, extend to meromorphic functions with poles on affine hyperplanes. We refine this result in showing that the poles lie on hyperplanes parallel to the facets of certain convex polyhedra associated to the defining matrix for the Shintani zeta function. Explicitly, the latter are the Newton polytopes of the polynomials induced by the columns of the underlying matrix. We then prove that the coefficients of the equation which describes the hyperplanes in the canonical basis are either zero or one, similar to the poles arising when renormalising generic Feynman amplitudes. For that purpose, we introduce an algorithm to distribute weight over a graph such that the weight at each vertex satisfies a given lower bound.