Residues and tame symbols on toroidal varieties

Abstract

We introduce a new approach to the study of a system of algebraic equations in $(\mathbb{C}^\times)^n$ 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.