Multiplicative intersection theory and complex tropical varieties

B. Ya. Kazarnovskii

Izvestiya. Mathematics , Tome 71 (2007) no. 4, pp. 673-720.

Résumé

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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