Abstract:
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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