Deciding eventual positivity of polynomials

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Abstract
Let P and f be polynomials in several (real) variables, with P having no negative coefficients. We give necessary and sufficient conditions for there to exist a positive integer n with Pnf having no negative coefficients; roughly speaking, the conditions involve the behaviour of f as a function on the positive orthant, together with its behaviour on a boundary constructed from the supporting monomials of P. This completes a series of results due to Poincaré (1883), Meissner (1911), and Polyà (1927). The former discusses the one variable case, the latter two deal with the situation that the Newton polyhedra of both P and f be, respectively, standard hypercubes, standard simplices.

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