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We study subvarieties in the torus, defined by equations $f = 0$ for generic $f \\in L$. We generalize the BKK theorem, that counts the number of solutions of a system of Laurent polynomial equations generic for their Newton polytopes, to this setting. The answer is in terms of mixed volume of certain virtual polytopes encoding discrete invariants of $L$ which involves matroid data. 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We study subvarieties in the torus, defined by equations $f = 0$ for generic $f \\in L$. We generalize the BKK theorem, that counts the number of solutions of a system of Laurent polynomial equations generic for their Newton polytopes, to this setting. The answer is in terms of mixed volume of certain virtual polytopes encoding discrete invariants of $L$ which involves matroid data. 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