We introduce a new approach to the study of systems of algebraic equations whose Newton polyhedra have sufficiently general relative locations, based on the theory of tame symbols and residues due to Parshin.;We give a new explicit description of combinatorial coefficients, which are geometric invariants that reflect the relative location of a collection of n convex compact polyhedra in Rn . Combinatorial coefficients are one of the main ingredients in Khovanskii's recent result on the product of the roots of a system of n algebraic equations in ( Cx )n whose Newton polyhedra have sufficiently general relative locations, and in the Gelfond-Khovanskii formula for the sum of the Grothendieck residues over the roots of such systems. Our description puts the combinatorial coefficient into the framework of Parshin's theory.;We consider Parshin's theory of residues and tame symbols on toroidal varieties. It turns out to be more explicit than the general theory, and it is enriched with the combinatorics inherited from toroidal varieties. Our description of the combinatorial coefficients is essential for the proof of our main results on residues and symbols on toroidal varieties. They provide a uniform explanation of both the Khovanskii and Gelfond-Khovanskii formulae in terms of the theory of symbols and residues on toroidal varieties, and extend them to the case of an algebraically closed field of arbitrary characteristic.
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