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A.Khovanskii

Parshin's symbols, toric geometry and product of the roots of a system of equations

According to the famous theorem of A.~Weil the product of so-called Weil's symbols $\{f,g\}$ over all the points of an algebraic curve $\Gamma$ is equal to 1. Here $f$, $g$ are non-zero meromorhpic functions on $\Gamma$. It turns out that one can obtain a very simple proof of this theorem just by looking at the Newton polygon of the equation of the image of the curve $\Gamma$ under the meromorphic map $f,g\:\Gamma \to (\Bbb C^*)^2$. Parshin generalized Weil's theorem to the multidimensional case and defined so-called Parshin's symbols of $(n+1)$ meromorphic functions on a $n$-dimensional variety. Pashin's construction is pure algebraic. I will present a new topological explanation of the Parshin theory and a multidimensional generalization of the classical Vieta's formula for the product of all the roots of a polynomial.