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It is known that Shintani zeta functions, which generalise multiple zetafunctions, extend to meromorphic functions with poles on affine hyperplanes. Werefine this result in showing that the poles lie on hyperplanes parallel to thefacets of certain convex polyhedra associated to the defining matrix for theShintani zeta function. Explicitly, the latter are the Newton polytopes of thepolynomials induced by the columns of the underlying matrix. We then prove thatthe coefficients of the equation which describes the hyperplanes in thecanonical basis are either zero or one, similar to the poles arising whenrenormalising generic Feynman amplitudes. For that purpose, we introduce analgorithm to distribute weight over a graph such that the weight at each vertexsatisfies a given lower bound.

Pole structure of Shintani zeta functions and Newton polytopes