Geodesic
Multiplicative intersection theory and complex tropical varieties
Izvestiya. Mathematics , Tome 71 (2007) no. 4, pp. 673-720

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We develop an intersection theory for subvarieties of a torus. Besides the number of intersection points for a generic pair of subvarieties of complementary dimensions, this theory takes into account the product of these points as elements of the ambient torus. In the case of a complete intersection of divisors, our intersection theory yields Bernshtein's formula for the number of roots of a system as well as Khovanskii's formula for their product. When constructing this theory, we naturally encounter ‘piecewise-linear’ subsets of the torus which are referred to as complex tropical varieties. 
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     author = {B. Ya. Kazarnovskii},
     title = {Multiplicative intersection theory and complex tropical varieties},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2007_71_4_a1/}
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B. Ya. Kazarnovskii. Multiplicative intersection theory and complex tropical varieties. Izvestiya. Mathematics , Tome 71 (2007) no. 4, pp. 673-720. http://geodesic.mathdoc.fr/item/IM2_2007_71_4_a1/