  We introduce a new approach to the study of a system of algebraic equations
in the algebraic torus whose Newton polytopes have sufficiently general
relative positions. Our method is based on the theory of Parshin's residues and
tame symbols on toroidal varieties. It provides a uniform algebraic explanation
of the recent result of Khovanskii on the product of the roots of such systems
and the Gel'fond--Khovanskii result on the sum of the values of a Laurent
polynomial over the roots of such systems, and extends them to the case of an
algebraically closed field of arbitrary characteristic.
