  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.
