In the preprint http://arxiv.org/pdf/0812.0433.pdf, Kaveh and Khovanskii define an intersection index for an n-tuple of finite-dimensional spaces of rational functions on an irreducible n-dimensional complex algebraic variety. Among the properties of the intersection index they prove, there is an analog of Alexandrov-Fenchel inequality (Theorem 4.28 and Corollary 4.29).

In the preprint Mixed volume and an extension of intersection theory of divisors, Kaveh and Khovanskii define an intersection index for an $n$-tuple of finite-dimensional spaces of rational functions on an irreducible $n$-dimensional complex algebraic variety. Among the properties of the intersection index they prove, there is an analog of Alexandrov–Fenchel inequality (Theorem 4.28 and Corollary 4.29).